Theorem or2expropbilem1 44413

Description: Lemma 1 for or2expropbi 44415 and ich2exprop 44811. (Contributed by AV, 16-Jul-2023.)

Assertion

Ref	Expression
or2expropbilem1	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑎,𝑏,𝑥,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝑋(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem or2expropbilem1

Step	Hyp	Ref	Expression
1		vex 3426	. . . . . . . 8 ⊢ 𝑎 ∈ V
2		vex 3426	. . . . . . . 8 ⊢ 𝑏 ∈ V
3	1, 2	pm3.2i 470	. . . . . . 7 ⊢ (𝑎 ∈ V ∧ 𝑏 ∈ V)
4	3	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑎 ∈ V ∧ 𝑏 ∈ V))
5	4	anim1ci 615	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) → (𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)))
6	5	adantr 480	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎 ∧ 𝐵 = 𝑏)) → (𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)))
7		sbcid 3728	. . . . . . 7 ⊢ ([𝑏 / 𝑏][𝑎 / 𝑎]𝜑 ↔ [𝑎 / 𝑎]𝜑)
8		sbcid 3728	. . . . . . 7 ⊢ ([𝑎 / 𝑎]𝜑 ↔ 𝜑)
9	7, 8	sylbbr 235	. . . . . 6 ⊢ (𝜑 → [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
10	9	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) → [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
11		opeq12 4803	. . . . 5 ⊢ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩)
12	10, 11	anim12ci 613	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎 ∧ 𝐵 = 𝑏)) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
13		nfv 1918	. . . . 5 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
14		nfv 1918	. . . . 5 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
15		opeq12 4803	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)
16	15	eqeq2d 2749	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
17		dfsbcq 3713	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑏][𝑥 / 𝑎]𝜑))
18		dfsbcq 3713	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ([𝑥 / 𝑎]𝜑 ↔ [𝑎 / 𝑎]𝜑))
19	18	sbcbidv 3770	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ([𝑏 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
20	17, 19	sylan9bbr 510	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
21	16, 20	anbi12d 630	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)))
22	21	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)))
23	13, 14, 22	spc2ed 3530	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
24	6, 12, 23	sylc 65	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎 ∧ 𝐵 = 𝑏)) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
25	24	exp31 419	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝜑 → ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
26	25	com23 86	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422  [wsbc 3711  ⟨cop 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565

This theorem is referenced by:  or2expropbi  44415  ich2exprop  44811