Theorem cbval2v 1911

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 1516	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1516	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1516	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1516	. 2 ⊢ Ⅎ𝑦𝜓
5		cbval2v.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbval2 1909	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by: (None)