Theorem ceqsex2v 2595

Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Hypotheses

Ref	Expression
ceqsex2v.1	⊢ 𝐴 ∈ V
ceqsex2v.2	⊢ 𝐵 ∈ V
ceqsex2v.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ceqsex2v.4	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ceqsex2v	⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥   𝜒,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)

Proof of Theorem ceqsex2v

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 ⊢ Ⅎ𝑥𝜓
2		nfv 1421	. 2 ⊢ Ⅎ𝑦𝜒
3		ceqsex2v.1	. 2 ⊢ 𝐴 ∈ V
4		ceqsex2v.2	. 2 ⊢ 𝐵 ∈ V
5		ceqsex2v.3	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6		ceqsex2v.4	. 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
7	1, 2, 3, 4, 5, 6	ceqsex2 2594	1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)

Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  ceqsex3v  2596  ceqsex4v  2597