Theorem ralbid 2475

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)

Hypotheses

Ref	Expression
ralbid.1	⊢ Ⅎ𝑥𝜑
ralbid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem ralbid

Step	Hyp	Ref	Expression
1		ralbid.1	. 2 ⊢ Ⅎ𝑥𝜑
2		ralbid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	ralbida 2471	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1460   ∈ wcel 2148  ∀wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460

This theorem is referenced by:  ralbidv  2477  sbcralt  3041  riota5f  5857  mkvprop  7158  lble  8906  ellimc3apf  14214  strcollnft  14821