Theorem biraldva 1215

Description: Formula-building rule for restricted universal quantifier (deduction rule).

Hypothesis

Ref	Expression
biraldva.1	⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))

Assertion

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

Distinct variable group(s):   φ,x

Proof of Theorem biraldva

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (φ → ∀xφ)
2		biraldva.1	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	1, 2	biralda 1213	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ A χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  ordunisssuc 2334  tfindsg2 2403  weinxp 2467  f1oweOLD 2944  isfinite2 3437  kmlem2 3581  iscard 3659  sup3 4511  indstr 4611  mdbr2 5728  dmdbr2 5733

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  ax-17 925

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

metamath.org