Theorem copsex2g 1903

Description: Implicit substitution inference for ordered pairs.

Hypothesis

Ref	Expression
copsex2g.1	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))

Assertion

Ref	Expression
copsex2g	⊢ ((A ∈ C ∧ B ∈ D) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))

Distinct variable group(s):   x,y,ψ   x,A,y   x,B,y

Proof of Theorem copsex2g

Step	Hyp	Ref	Expression
1		eeanv 980	. . 3 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
2		hbe1 709	. . . . 5 ⊢ (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) → ∀x∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ))
3		ax-17 925	. . . . 5 ⊢ (ψ → ∀xψ)
4	2, 3	hbbi 705	. . . 4 ⊢ ((∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ) → ∀x(∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))
5		hbe1 709	. . . . . . 7 ⊢ (∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) → ∀y∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ))
6	5	hbex 701	. . . . . 6 ⊢ (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) → ∀y∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ))
7		ax-17 925	. . . . . 6 ⊢ (ψ → ∀yψ)
8	6, 7	hbbi 705	. . . . 5 ⊢ ((∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ) → ∀y(∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))
9		opeq12 1878	. . . . . . 7 ⊢ ((x = A ∧ y = B) → ⟨x, y⟩ = ⟨A, B⟩)
10		copsexg 1902	. . . . . . . 8 ⊢ (⟨A, B⟩ = ⟨x, y⟩ → (φ ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)))
11	10	cleqcoms 1104	. . . . . . 7 ⊢ (⟨x, y⟩ = ⟨A, B⟩ → (φ ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)))
12	9, 11	syl 12	. . . . . 6 ⊢ ((x = A ∧ y = B) → (φ ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)))
13		copsex2g.1	. . . . . 6 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
14	12, 13	bitr3d 408	. . . . 5 ⊢ ((x = A ∧ y = B) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))
15	8, 14	19.23ai 746	. . . 4 ⊢ (∃y(x = A ∧ y = B) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))
16	4, 15	19.23ai 746	. . 3 ⊢ (∃x∃y(x = A ∧ y = B) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))
17	1, 16	sylbir 176	. 2 ⊢ ((∃x x = A ∧ ∃y y = B) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))
18		elex 1356	. 2 ⊢ (A ∈ C → ∃x x = A)
19		elex 1356	. 2 ⊢ (B ∈ D → ∃y y = B)
20	17, 18, 19	syl2an 349	1 ⊢ ((A ∈ C ∧ B ∈ D) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810

This theorem is referenced by:  ltresr 4052

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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