Theorem cbvraldva 3251

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 2159, ax-ext 2741. (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 463	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝜓 ↔ 𝜒))
3	2	pm5.74da 815	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
4	3	cbvralvw 3249	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))
5		r19.21v 3196	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
6		r19.21v 3196	. . 3 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦 ∈ 𝐴 𝜒))
7	4, 5, 6	3bitr3i 304	. 2 ⊢ ((𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∀𝑦 ∈ 𝐴 𝜒))
8	7	pm5.74ri 275	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-clel 2844  df-ral 3086

This theorem is referenced by:  cbvrexdva  3252  wrd2ind  14756  axtgcont  28700  cbviindavw  36660  cbvixpdavw  36675  weiunfrlem  36860