Theorem cbvaldvaw 2037

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 2417 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) Reduce axiom usage, along an idea of GG. (Revised by Wolf Lammen, 10-Feb-2024.)

Hypothesis

Ref	Expression
cbvaldvaw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

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

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

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

Proof of Theorem cbvaldvaw

Step	Hyp	Ref	Expression
1		cbvaldvaw.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2	1	ancoms 458	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝜓 ↔ 𝜒))
3	2	pm5.74da 803	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
4	3	cbvalvw 2035	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒))
5		19.21v 1938	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
6		19.21v 1938	. . 3 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒))
7	4, 5, 6	3bitr3i 301	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒))
8	7	pm5.74ri 272	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778

This theorem is referenced by:  cbvexdvaw  2038  cbval2vw  2039  cbvmodavw  36216  cbvsbdavw  36220  cbvsbdavw2  36221  scottabf  44209  ismnu  44230