Theorem or2expropbilem1 43316

Description: Lemma 1 for or2expropbi 43318 and ich2exprop 43682. (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 3497	. . . . . . . 8 ⊢ 𝑎 ∈ V
2		vex 3497	. . . . . . . 8 ⊢ 𝑏 ∈ V
3	1, 2	pm3.2i 473	. . . . . . 7 ⊢ (𝑎 ∈ V ∧ 𝑏 ∈ V)
4	3	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑎 ∈ V ∧ 𝑏 ∈ V))
5	4	anim1ci 617	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) → (𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)))
6	5	adantr 483	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎 ∧ 𝐵 = 𝑏)) → (𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)))
7		sbcid 3789	. . . . . . 7 ⊢ ([𝑏 / 𝑏][𝑎 / 𝑎]𝜑 ↔ [𝑎 / 𝑎]𝜑)
8		sbcid 3789	. . . . . . 7 ⊢ ([𝑎 / 𝑎]𝜑 ↔ 𝜑)
9	7, 8	sylbbr 238	. . . . . 6 ⊢ (𝜑 → [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
10	9	adantl 484	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) → [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
11		opeq12 4805	. . . . 5 ⊢ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩)
12	10, 11	anim12ci 615	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎 ∧ 𝐵 = 𝑏)) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
13		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
14		nfv 1915	. . . . 5 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
15		opeq12 4805	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)
16	15	eqeq2d 2832	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
17		dfsbcq 3774	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑏][𝑥 / 𝑎]𝜑))
18		dfsbcq 3774	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ([𝑥 / 𝑎]𝜑 ↔ [𝑎 / 𝑎]𝜑))
19	18	sbcbidv 3827	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ([𝑏 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
20	17, 19	sylan9bbr 513	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
21	16, 20	anbi12d 632	. . . . . 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 484	. . . . 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 3602	. . . 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 422	. 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 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3494  [wsbc 3772  ⟨cop 4573

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

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  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-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574

This theorem is referenced by:  or2expropbi  43318  ich2exprop  43682