Theorem cbvraldva 3383

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.)

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	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2		eqidd 2739	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐴)
3	1, 2	cbvraldva2 3381	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-cleq 2730  df-clel 2817  df-ral 3068

This theorem is referenced by:  wrd2ind  14364  axtgcont  26734