Theorem cbvex2v 1936

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Hypothesis

Ref	Expression
cbval2v.1	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvex2v	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Distinct variable groups:   𝑧,𝑤,𝜑   𝑥,𝑦,𝜓   𝑥,𝑤   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑧,𝑤)

Proof of Theorem cbvex2v

Step	Hyp	Ref	Expression
1		nfv 1539	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1539	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1539	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1539	. 2 ⊢ Ⅎ𝑦𝜓
5		cbval2v.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvex2 1934	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1503

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  cbvex4v  1942  th3qlem1  6663