Theorem ralbid 2633

Description: Formula-building rule for restricted universal quantifier (deduction rule). (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 451	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
4	1, 3	ralbida 2629	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ A χ))

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  ralbidv  2635  sbcralt  3119  sbcrext  3120