Theorem optocl 5763

Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)

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 groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem optocl

Step	Hyp	Ref	Expression
1		elxp3 5735	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
2		opelxp 5705	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
3		optocl.3	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)
4	2, 3	sylbi 216	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜑)
5		optocl.2	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))
6	4, 5	imbitrid 243	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜓))
7	6	imp 406	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
8	7	exlimivv 1927	. . 3 ⊢ (∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
9	1, 8	sylbi 216	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝜓)
10		optocl.1	. 2 ⊢ 𝐷 = (𝐵 × 𝐶)
11	9, 10	eleq2s 2845	1 ⊢ (𝐴 ∈ 𝐷 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ⟨cop 4629   × cxp 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-xp 5675

This theorem is referenced by:  2optocl  5764  3optocl  5765  ecoptocl  8800  ax1rid  11155  axcnre  11158