Theorem impprop 38558

Description: The implication between two sentences of propositional calculus is a sentence of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)

Assertion

Ref	Expression
impprop	⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → (𝑥prop→𝑦) ∈ PROP)

Proof of Theorem impprop

Dummy variables 𝑤 𝑢 𝑣 𝑧 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prop 38554	. . 3 ⊢ PROP = setrecs((𝑤 ∈ V ↦ {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))}))
2		prex 5396	. . . 4 ⊢ {𝑥, 𝑦} ∈ V
3	2	a1i 11	. . 3 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → {𝑥, 𝑦} ∈ V)
4		prssi 4782	. . 3 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → {𝑥, 𝑦} ⊆ PROP)
5	1, 3, 4	setrec1 9929	. 2 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → ((𝑤 ∈ V ↦ {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))})‘{𝑥, 𝑦}) ⊆ PROP)
6		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
7		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
8		oveq2 7417	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑢prop→𝑣) = (𝑢prop→𝑦))
9	8	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → (𝑧 = (𝑢prop→𝑣) ↔ 𝑧 = (𝑢prop→𝑦)))
10		oveq1 7416	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢prop→𝑦) = (𝑥prop→𝑦))
11	10	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → (𝑧 = (𝑢prop→𝑦) ↔ 𝑧 = (𝑥prop→𝑦)))
12	6, 7, 9, 11	ceqsex2v 3501	. . . . . . . . . . 11 ⊢ (∃𝑣∃𝑢(𝑣 = 𝑦 ∧ 𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) ↔ 𝑧 = (𝑥prop→𝑦))
13	12	bilanri 512	. . . . . . . . . 10 ⊢ (((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) ∧ 𝑧 = (𝑥prop→𝑦)) → ∃𝑣∃𝑢(𝑣 = 𝑦 ∧ 𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)))
14		3anass 1111	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑦 ∧ 𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) ↔ (𝑣 = 𝑦 ∧ (𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))))
15	14	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑢(𝑣 = 𝑦 ∧ 𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) ↔ ∃𝑢(𝑣 = 𝑦 ∧ (𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))))
16		19.42v 1986	. . . . . . . . . . . 12 ⊢ (∃𝑢(𝑣 = 𝑦 ∧ (𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))) ↔ (𝑣 = 𝑦 ∧ ∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))))
17	15, 16	bitri 278	. . . . . . . . . . 11 ⊢ (∃𝑢(𝑣 = 𝑦 ∧ 𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) ↔ (𝑣 = 𝑦 ∧ ∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))))
18	17	exbii 1881	. . . . . . . . . 10 ⊢ (∃𝑣∃𝑢(𝑣 = 𝑦 ∧ 𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) ↔ ∃𝑣(𝑣 = 𝑦 ∧ ∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))))
19	13, 18	sylib 221	. . . . . . . . 9 ⊢ (((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) ∧ 𝑧 = (𝑥prop→𝑦)) → ∃𝑣(𝑣 = 𝑦 ∧ ∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))))
20		olc 882	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → (𝑣 = 𝑥 ∨ 𝑣 = 𝑦))
21		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
22	21	elpr 4609	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝑥, 𝑦} ↔ (𝑣 = 𝑥 ∨ 𝑣 = 𝑦))
23	20, 22	sylibr 237	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑦 → 𝑣 ∈ {𝑥, 𝑦})
24		orc 881	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 → (𝑢 = 𝑥 ∨ 𝑢 = 𝑦))
25		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑢 ∈ V
26	25	elpr 4609	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {𝑥, 𝑦} ↔ (𝑢 = 𝑥 ∨ 𝑢 = 𝑦))
27	24, 26	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑥 → 𝑢 ∈ {𝑥, 𝑦})
28	27	anim1i 627	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) → (𝑢 ∈ {𝑥, 𝑦} ∧ 𝑧 = (𝑢prop→𝑣)))
29	28	eximi 1868	. . . . . . . . . . . . 13 ⊢ (∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) → ∃𝑢(𝑢 ∈ {𝑥, 𝑦} ∧ 𝑧 = (𝑢prop→𝑣)))
30		df-rex 3087	. . . . . . . . . . . . 13 ⊢ (∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣) ↔ ∃𝑢(𝑢 ∈ {𝑥, 𝑦} ∧ 𝑧 = (𝑢prop→𝑣)))
31	29, 30	sylibr 237	. . . . . . . . . . . 12 ⊢ (∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) → ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣))
32	31	olcd 888	. . . . . . . . . . 11 ⊢ (∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) → (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)))
33	23, 32	anim12i 625	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑦 ∧ ∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))) → (𝑣 ∈ {𝑥, 𝑦} ∧ (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣))))
34	33	eximi 1868	. . . . . . . . 9 ⊢ (∃𝑣(𝑣 = 𝑦 ∧ ∃𝑢(𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))) → ∃𝑣(𝑣 ∈ {𝑥, 𝑦} ∧ (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣))))
35	19, 34	syl 18	. . . . . . . 8 ⊢ (((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) ∧ 𝑧 = (𝑥prop→𝑦)) → ∃𝑣(𝑣 ∈ {𝑥, 𝑦} ∧ (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣))))
36		df-rex 3087	. . . . . . . 8 ⊢ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ↔ ∃𝑣(𝑣 ∈ {𝑥, 𝑦} ∧ (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣))))
37	35, 36	sylibr 237	. . . . . . 7 ⊢ (((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) ∧ 𝑧 = (𝑥prop→𝑦)) → ∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)))
38	37	orcd 887	. . . . . 6 ⊢ (((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) ∧ 𝑧 = (𝑥prop→𝑦)) → (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛)))
39	38	ex 418	. . . . 5 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → (𝑧 = (𝑥prop→𝑦) → (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))))
40	39	alrimiv 1960	. . . 4 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → ∀𝑧(𝑧 = (𝑥prop→𝑦) → (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))))
41		ovex 7442	. . . . 5 ⊢ (𝑥prop→𝑦) ∈ V
42		elab6g 3623	. . . . 5 ⊢ ((𝑥prop→𝑦) ∈ V → ((𝑥prop→𝑦) ∈ {𝑧 ∣ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))} ↔ ∀𝑧(𝑧 = (𝑥prop→𝑦) → (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛)))))
43	41, 42	ax-mp 5	. . . 4 ⊢ ((𝑥prop→𝑦) ∈ {𝑧 ∣ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))} ↔ ∀𝑧(𝑧 = (𝑥prop→𝑦) → (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))))
44	40, 43	sylibr 237	. . 3 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → (𝑥prop→𝑦) ∈ {𝑧 ∣ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))})
45		rexeq 3315	. . . . . . . . 9 ⊢ (𝑤 = {𝑥, 𝑦} → (∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣) ↔ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)))
46	45	orbi2d 929	. . . . . . . 8 ⊢ (𝑤 = {𝑥, 𝑦} → ((𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ↔ (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣))))
47	46	rexeqbi1dv 3330	. . . . . . 7 ⊢ (𝑤 = {𝑥, 𝑦} → (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ↔ ∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣))))
48	47	orbi1d 930	. . . . . 6 ⊢ (𝑤 = {𝑥, 𝑦} → ((∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛)) ↔ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))))
49	48	abbidv 2826	. . . . 5 ⊢ (𝑤 = {𝑥, 𝑦} → {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))} = {𝑧 ∣ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))})
50		eqid 2760	. . . . 5 ⊢ (𝑤 ∈ V ↦ {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))}) = (𝑤 ∈ V ↦ {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))})
51	2	dfproplem 38555	. . . . 5 ⊢ {𝑧 ∣ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))} ∈ V
52	49, 50, 51	fvmpt 6982	. . . 4 ⊢ ({𝑥, 𝑦} ∈ V → ((𝑤 ∈ V ↦ {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))})‘{𝑥, 𝑦}) = {𝑧 ∣ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))})
53	2, 52	ax-mp 5	. . 3 ⊢ ((𝑤 ∈ V ↦ {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))})‘{𝑥, 𝑦}) = {𝑧 ∣ (∃𝑣 ∈ {𝑥, 𝑦} (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ {𝑥, 𝑦}𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))}
54	44, 53	eleqtrrdi 2871	. 2 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → (𝑥prop→𝑦) ∈ ((𝑤 ∈ V ↦ {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))})‘{𝑥, 𝑦}))
55	5, 54	sseldd 3932	1 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → (𝑥prop→𝑦) ∈ PROP)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∃wrex 3086  Vcvv 3450  {cpr 4586   ↦ cmpt 5186  ‘cfv 6528  (class class class)co 7409  ℕcn 12290  propvar cpropvar 38547  prop¬cpropneg 38548  prop→cpropimp 38549  PROPcprop 38553

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-reg 9564  ax-inf2 9620  ax-cnex 11213  ax-1cn 11215  ax-addcl 11217

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-om 7862  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-r1 9746  df-rank 9747  df-scott 9886  df-setrecs 9922  df-nn 12291  df-prop 38554

This theorem is used by:  dfprop1  38559