Theorem sbcpr 43732

Description: The proper substitution of an unordered pair for a setvar variable corresponds to a proper substitution of each of its elements. (Contributed by AV, 7-Apr-2023.)

Hypothesis

Ref	Expression
sbcpr.x	⊢ (𝑝 = {𝑥, 𝑦} → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcpr	⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)

Distinct variable groups:   𝑎,𝑏,𝑥,𝑦,𝑝   𝜑,𝑥,𝑦   𝜓,𝑝

Allowed substitution hints:   𝜑(𝑝,𝑎,𝑏)   𝜓(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem sbcpr

Step	Hyp	Ref	Expression
1		sbc5 3800	. . 3 ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ ∃𝑝(𝑝 = {𝑎, 𝑏} ∧ 𝜑))
2		preq12 4671	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
3	2	eqcomd 2827	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑎, 𝑏} = {𝑥, 𝑦})
4	3	eqeq2d 2832	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑝 = {𝑎, 𝑏} ↔ 𝑝 = {𝑥, 𝑦}))
5	4	biimpa 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑥, 𝑦})
6		sbcpr.x	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {𝑥, 𝑦} → (𝜑 ↔ 𝜓))
7	5, 6	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜑 ↔ 𝜓))
8	7	biimpd 231	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜑 → 𝜓))
9	8	expcom 416	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜑 → 𝜓)))
10	9	expd 418	. . . . . . . . . . 11 ⊢ (𝑝 = {𝑎, 𝑏} → (𝑥 = 𝑎 → (𝑦 = 𝑏 → (𝜑 → 𝜓))))
11	10	com24 95	. . . . . . . . . 10 ⊢ (𝑝 = {𝑎, 𝑏} → (𝜑 → (𝑦 = 𝑏 → (𝑥 = 𝑎 → 𝜓))))
12	11	imp31 420	. . . . . . . . 9 ⊢ (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → (𝑥 = 𝑎 → 𝜓))
13	12	alrimiv 1928	. . . . . . . 8 ⊢ (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → ∀𝑥(𝑥 = 𝑎 → 𝜓))
14		vex 3497	. . . . . . . . 9 ⊢ 𝑎 ∈ V
15	14	sbc6 3802	. . . . . . . 8 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑎 → 𝜓))
16	13, 15	sylibr 236	. . . . . . 7 ⊢ (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → [𝑎 / 𝑥]𝜓)
17	16	ex 415	. . . . . 6 ⊢ ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → (𝑦 = 𝑏 → [𝑎 / 𝑥]𝜓))
18	17	alrimiv 1928	. . . . 5 ⊢ ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → ∀𝑦(𝑦 = 𝑏 → [𝑎 / 𝑥]𝜓))
19		vex 3497	. . . . . 6 ⊢ 𝑏 ∈ V
20	19	sbc6 3802	. . . . 5 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ ∀𝑦(𝑦 = 𝑏 → [𝑎 / 𝑥]𝜓))
21	18, 20	sylibr 236	. . . 4 ⊢ ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
22	21	exlimiv 1931	. . 3 ⊢ (∃𝑝(𝑝 = {𝑎, 𝑏} ∧ 𝜑) → [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
23	1, 22	sylbi 219	. 2 ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 → [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
24		sbc5 3800	. . . . 5 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ ∃𝑦(𝑦 = 𝑏 ∧ [𝑎 / 𝑥]𝜓))
25		sbc5 3800	. . . . . . . 8 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑎 ∧ 𝜓))
26	6	bicomd 225	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {𝑥, 𝑦} → (𝜓 ↔ 𝜑))
27	5, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜓 ↔ 𝜑))
28	27	biimpd 231	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜓 → 𝜑))
29	28	expcom 416	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜓 → 𝜑)))
30	29	com13 88	. . . . . . . . . . 11 ⊢ (𝜓 → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑝 = {𝑎, 𝑏} → 𝜑)))
31	30	expd 418	. . . . . . . . . 10 ⊢ (𝜓 → (𝑥 = 𝑎 → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑))))
32	31	impcom 410	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝜓) → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
33	32	exlimiv 1931	. . . . . . . 8 ⊢ (∃𝑥(𝑥 = 𝑎 ∧ 𝜓) → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
34	25, 33	sylbi 219	. . . . . . 7 ⊢ ([𝑎 / 𝑥]𝜓 → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
35	34	impcom 410	. . . . . 6 ⊢ ((𝑦 = 𝑏 ∧ [𝑎 / 𝑥]𝜓) → (𝑝 = {𝑎, 𝑏} → 𝜑))
36	35	exlimiv 1931	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝑏 ∧ [𝑎 / 𝑥]𝜓) → (𝑝 = {𝑎, 𝑏} → 𝜑))
37	24, 36	sylbi 219	. . . 4 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 → (𝑝 = {𝑎, 𝑏} → 𝜑))
38	37	alrimiv 1928	. . 3 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 → ∀𝑝(𝑝 = {𝑎, 𝑏} → 𝜑))
39		prex 5333	. . . 4 ⊢ {𝑎, 𝑏} ∈ V
40	39	sbc6 3802	. . 3 ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ ∀𝑝(𝑝 = {𝑎, 𝑏} → 𝜑))
41	38, 40	sylibr 236	. 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 → [{𝑎, 𝑏} / 𝑝]𝜑)
42	23, 41	impbii 211	1 ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780  [wsbc 3772  {cpr 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-nul 4292  df-sn 4568  df-pr 4570

This theorem is referenced by:  reupr  43733