Theorem cbvalv 2391

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2363. See cbvalvw 2031 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2129 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cbvalv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvalv	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem cbvalv

Step	Hyp	Ref	Expression
1		nfv 1909	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1909	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvalv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbval 2389	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-11 2146  ax-12 2163  ax-13 2363

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778

This theorem is referenced by:  cbval2vv  2404  cdeqal1  3760