Theorem albid 1772

Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
albid	⊢ (φ → (∀xψ ↔ ∀xχ))

Proof of Theorem albid

Step	Hyp	Ref	Expression
1		albid.1	. . 3 ⊢ Ⅎxφ
2	1	nfri 1762	. 2 ⊢ (φ → ∀xφ)
3		albid.2	. 2 ⊢ (φ → (ψ ↔ χ))
4	2, 3	albidh 1590	1 ⊢ (φ → (∀xψ ↔ ∀xχ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544

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-ex 1542  df-nf 1545

This theorem is referenced by:  nfbidf  1774  ax11v2  1992  sbcom  2089  sbal2  2134  ax11eq  2193  ax11el  2194  ax11v2-o  2201  eubid  2211  ralbida  2629  raleqf  2804  intab  3957