Theorem cbvraldva 3153

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 2611	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐴)
3	1, 2	cbvraldva2 3151	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-ral 2901

This theorem is referenced by:  wrd2ind  13329  axtgcont  25168