Theorem ralbid 1664

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

Hypotheses

Ref	Expression
ralbid.1	⊢ (φ → ∀xφ)
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 ⊢ (φ → ∀xφ)
2		ralbid.2	. . 3 ⊢ (φ → (ψ ↔ χ))
3	2	adantr 391	. 2 ⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))
4	1, 3	ralbida 1660	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ A χ))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 956   ∈ wcel 960  ∀wral 1648

This theorem is referenced by:  ralbidv 1666  ralbii 1670  sbcralt 1993  sbcrext 1994  sbcralgf 1995  sbcrexgf 1996  zfrep6 3620  cplem2 4731  ac6lem 4764  lble 6049  irredt 10317

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1652

Copyright terms: Public domain