Theorem cbvaldw 2347

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvald 2417 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 2-Jan-2002.) (Revised by Gino Giotto, 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 1915	. 2 ⊢ Ⅎ𝑥𝜑
2		cbvaldw.1	. 2 ⊢ Ⅎ𝑦𝜑
3		cbvaldw.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfvd 1916	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbvaldw.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	1, 2, 3, 4, 5	cbv2w 2346	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  cbvexdw  2348