Theorem optocl 2469

Description: Implicit substitution of class for ordered pair.

Hypotheses

Ref	Expression
optocl.1	⊢ D = (B × C)
optocl.2	⊢ (⟨x, y⟩ = A → (φ ↔ ψ))
optocl.3	⊢ ((x ∈ B ∧ y ∈ C) → φ)

Assertion

Ref	Expression
optocl	⊢ (A ∈ D → ψ)

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

Proof of Theorem optocl

Step	Hyp	Ref	Expression
1		optocl.1	. . 3 ⊢ D = (B × C)
2	1	eleq2i 1153	. 2 ⊢ (A ∈ D ↔ A ∈ (B × C))
3		elxp3 2460	. . 3 ⊢ (A ∈ (B × C) ↔ ∃x∃y(⟨x, y⟩ = A ∧ ⟨x, y⟩ ∈ (B × C)))
4		optocl.2	. . . . . 6 ⊢ (⟨x, y⟩ = A → (φ ↔ ψ))
5		visset 1350	. . . . . . . 8 ⊢ y ∈ V
6	5	opelxp 2452	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (B × C) ↔ (x ∈ B ∧ y ∈ C))
7		optocl.3	. . . . . . 7 ⊢ ((x ∈ B ∧ y ∈ C) → φ)
8	6, 7	sylbi 174	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (B × C) → φ)
9	4, 8	syl5bi 183	. . . . 5 ⊢ (⟨x, y⟩ = A → (⟨x, y⟩ ∈ (B × C) → ψ))
10	9	imp 277	. . . 4 ⊢ ((⟨x, y⟩ = A ∧ ⟨x, y⟩ ∈ (B × C)) → ψ)
11	10	19.23aivv 953	. . 3 ⊢ (∃x∃y(⟨x, y⟩ = A ∧ ⟨x, y⟩ ∈ (B × C)) → ψ)
12	3, 11	sylbi 174	. 2 ⊢ (A ∈ (B × C) → ψ)
13	2, 12	sylbi 174	1 ⊢ (A ∈ D → ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  2optocl 2470  3optocl 2471  ecoptocl 3239  ax0id 4076  ax1id 4077  axnegex 4078  axrecex 4079  axcnre 4087

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  df-opab 2098  df-xp 2424

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