Theorem impprop 38645

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 38641	. . 3 ⊢ PROP = setrecs((𝑤 ∈ V ↦ {𝑧 ∣ (∃𝑣 ∈ 𝑤 (𝑧 = (prop¬‘𝑣) ∨ ∃𝑢 ∈ 𝑤 𝑧 = (𝑢prop→𝑣)) ∨ ∃𝑛 ∈ ℕ 𝑧 = (propvar ‘𝑛))}))
2		prex 5395	. . . 4 ⊢ {𝑥, 𝑦} ∈ V
3	2	a1i 11	. . 3 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → {𝑥, 𝑦} ∈ V)
4		prssi 4781	. . 3 ⊢ ((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → {𝑥, 𝑦} ⊆ PROP)
5	1, 3, 4	setrec1 9972	. 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 7428	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑢prop→𝑣) = (𝑢prop→𝑦))
9	8	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → (𝑧 = (𝑢prop→𝑣) ↔ 𝑧 = (𝑢prop→𝑦)))
10		oveq1 7427	. . . . . . . . . . . . 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 1880	. . . . . . . . . . . 12 ⊢ (∃𝑢(𝑣 = 𝑦 ∧ 𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣)) ↔ ∃𝑢(𝑣 = 𝑦 ∧ (𝑢 = 𝑥 ∧ 𝑧 = (𝑢prop→𝑣))))
16		19.42v 1985	. . . . . . . . . . . 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 1880	. . . . . . . . . 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 4608	. . . . . . . . . . . 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 4608	. . . . . . . . . . . . . . . 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 1867	. . . . . . . . . . . . 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 1867	. . . . . . . . 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 1959	. . . 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 7453	. . . . 5 ⊢ (𝑥prop→𝑦) ∈ V
42		elab6g 3622	. . . . 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 38642	. . . . 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 6993	. . . 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 3909	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 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450  {cpr 4585   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420  ℕcn 12335  propvar cpropvar 38634  prop¬cpropneg 38635  prop→cpropimp 38636  PROPcprop 38640

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642  ax-cnex 11256  ax-1cn 11258  ax-addcl 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769  df-scott 9929  df-setrecs 9965  df-nn 12336  df-prop 38641

This theorem is used by:  dfprop1  38646