Theorem cbval2v 1798

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)

Hypothesis

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

Assertion

Ref	Expression
cbval2v	⊢ (∀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 cbval2v

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1421	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1421	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1421	. 2 ⊢ Ⅎ𝑦𝜓
5		cbval2v.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbval2 1796	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241

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-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by: (None)