Theorem ceqsex2v 2814

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 1551	. 2 ⊢ Ⅎ𝑥𝜓
2		nfv 1551	. 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 2813	1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 981   = wceq 1373  ∃wex 1515   ∈ wcel 2176  Vcvv 2772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  ceqsex3v  2815  ceqsex4v  2816