Theorem sbcpr 47508

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 3816	. . 3 ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ ∃𝑝(𝑝 = {𝑎, 𝑏} ∧ 𝜑))
2		preq12 4735	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
3	2	eqcomd 2743	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑎, 𝑏} = {𝑥, 𝑦})
4	3	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑝 = {𝑎, 𝑏} ↔ 𝑝 = {𝑥, 𝑦}))
5	4	biimpa 476	. . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜑 → 𝜓))
9	8	expcom 413	. . . . . . . . . . . 12 ⊢ (𝑝 = {𝑎, 𝑏} → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜑 → 𝜓)))
10	9	expd 415	. . . . . . . . . . 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 417	. . . . . . . . 9 ⊢ (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → (𝑥 = 𝑎 → 𝜓))
13	12	alrimiv 1927	. . . . . . . 8 ⊢ (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → ∀𝑥(𝑥 = 𝑎 → 𝜓))
14		vex 3484	. . . . . . . . 9 ⊢ 𝑎 ∈ V
15	14	sbc6 3819	. . . . . . . 8 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑎 → 𝜓))
16	13, 15	sylibr 234	. . . . . . 7 ⊢ (((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ∧ 𝑦 = 𝑏) → [𝑎 / 𝑥]𝜓)
17	16	ex 412	. . . . . 6 ⊢ ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → (𝑦 = 𝑏 → [𝑎 / 𝑥]𝜓))
18	17	alrimiv 1927	. . . . 5 ⊢ ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → ∀𝑦(𝑦 = 𝑏 → [𝑎 / 𝑥]𝜓))
19		vex 3484	. . . . . 6 ⊢ 𝑏 ∈ V
20	19	sbc6 3819	. . . . 5 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ ∀𝑦(𝑦 = 𝑏 → [𝑎 / 𝑥]𝜓))
21	18, 20	sylibr 234	. . . 4 ⊢ ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) → [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
22	21	exlimiv 1930	. . 3 ⊢ (∃𝑝(𝑝 = {𝑎, 𝑏} ∧ 𝜑) → [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
23	1, 22	sylbi 217	. 2 ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 → [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
24		sbc5 3816	. . . . 5 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ ∃𝑦(𝑦 = 𝑏 ∧ [𝑎 / 𝑥]𝜓))
25		sbc5 3816	. . . . . . . 8 ⊢ ([𝑎 / 𝑥]𝜓 ↔ ∃𝑥(𝑥 = 𝑎 ∧ 𝜓))
26	6	bicomd 223	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {𝑥, 𝑦} → (𝜓 ↔ 𝜑))
27	5, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜓 ↔ 𝜑))
28	27	biimpd 229	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜓 → 𝜑))
29	28	expcom 413	. . . . . . . . . . . 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 415	. . . . . . . . . 10 ⊢ (𝜓 → (𝑥 = 𝑎 → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑))))
32	31	impcom 407	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝜓) → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
33	32	exlimiv 1930	. . . . . . . 8 ⊢ (∃𝑥(𝑥 = 𝑎 ∧ 𝜓) → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
34	25, 33	sylbi 217	. . . . . . 7 ⊢ ([𝑎 / 𝑥]𝜓 → (𝑦 = 𝑏 → (𝑝 = {𝑎, 𝑏} → 𝜑)))
35	34	impcom 407	. . . . . 6 ⊢ ((𝑦 = 𝑏 ∧ [𝑎 / 𝑥]𝜓) → (𝑝 = {𝑎, 𝑏} → 𝜑))
36	35	exlimiv 1930	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝑏 ∧ [𝑎 / 𝑥]𝜓) → (𝑝 = {𝑎, 𝑏} → 𝜑))
37	24, 36	sylbi 217	. . . 4 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 → (𝑝 = {𝑎, 𝑏} → 𝜑))
38	37	alrimiv 1927	. . 3 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 → ∀𝑝(𝑝 = {𝑎, 𝑏} → 𝜑))
39		prex 5437	. . . 4 ⊢ {𝑎, 𝑏} ∈ V
40	39	sbc6 3819	. . 3 ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ ∀𝑝(𝑝 = {𝑎, 𝑏} → 𝜑))
41	38, 40	sylibr 234	. 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 → [{𝑎, 𝑏} / 𝑝]𝜑)
42	23, 41	impbii 209	1 ⊢ ([{𝑎, 𝑏} / 𝑝]𝜑 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779  [wsbc 3788  {cpr 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629

This theorem is referenced by:  reupr  47509