Theorem cbvaldw 2346

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvald 2415 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvaldw.1	⊢ Ⅎ𝑦𝜑
cbvaldw.2	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbvaldw.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbvaldw	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥   𝑥,𝑦

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

Proof of Theorem cbvaldw

Step	Hyp	Ref	Expression
1		nfv 1921	. 2 ⊢ Ⅎ𝑥𝜑
2		cbvaldw.1	. 2 ⊢ Ⅎ𝑦𝜑
3		cbvaldw.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfvd 1922	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbvaldw.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	1, 2, 3, 4, 5	cbv2w 2345	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  cbvexdw  2347  axtcond  36706  mh-setindnd  36765  wl-sb8eutv  37950