Theorem cbvraldva 3237

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Avoid ax-9 2117, ax-ext 2704. (Revised by Wolf Lammen, 9-Mar-2025.)

Hypothesis

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

Assertion

Ref	Expression
cbvraldva	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

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

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

Proof of Theorem cbvraldva

Step	Hyp	Ref	Expression
1		cbvraldva.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2	1	ancoms 460	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝜓 ↔ 𝜒))
3	2	pm5.74da 803	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
4	3	cbvralvw 3235	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))
5		r19.21v 3180	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
6		r19.21v 3180	. . 3 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦 ∈ 𝐴 𝜒))
7	4, 5, 6	3bitr3i 301	. 2 ⊢ ((𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∀𝑦 ∈ 𝐴 𝜒))
8	7	pm5.74ri 272	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wral 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2811  df-ral 3063

This theorem is referenced by:  cbvrexdva  3238  wrd2ind  14673  axtgcont  27720