Theorem birala 1228

Description: Inference adding restricted universal quantifier to both sides of an equivalence.

Hypothesis

Ref	Expression
birala.1	⊢ (x ∈ A → (φ ↔ ψ))

Assertion

Ref	Expression
birala	⊢ (∀x ∈ A φ ↔ ∀x ∈ A ψ)

Proof of Theorem birala

Step	Hyp
Ref	Expression
1		birala.1	. . . 4 ⊢ (x ∈ A &rarÜ; (φ ↔ ψ))
2	1	pm5.74i 443	. . 3 ⊢ ((x ∈ A → φ) ↔ (x ∈ A → ψ))
3	2	bial 695	. 2 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ A → ψ))
4		df-ral 1205	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
5		df-ral 1205	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
6	3, 4, 5	3bitr4 158	1 ⊢ (∀x ∈ A φ ↔ ∀x ∈ A ψ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  fvreseq 2882  aceq4 3557  hods 5606  large 5700  elat2 5739

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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