Theorem albiim 1611

Description: Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)

Assertion

Ref	Expression
albiim	⊢ (∀x(φ ↔ ψ) ↔ (∀x(φ → ψ) ∧ ∀x(ψ → φ)))

Proof of Theorem albiim

Step	Hyp	Ref	Expression
1		dfbi2 609	. . 3 ⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ)))
2	1	albii 1566	. 2 ⊢ (∀x(φ ↔ ψ) ↔ ∀x((φ → ψ) ∧ (ψ → φ)))
3		19.26 1593	. 2 ⊢ (∀x((φ → ψ) ∧ (ψ → φ)) ↔ (∀x(φ → ψ) ∧ ∀x(ψ → φ)))
4	2, 3	bitri 240	1 ⊢ (∀x(φ ↔ ψ) ↔ (∀x(φ → ψ) ∧ ∀x(ψ → φ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  2albiim  1612  equveli  1988  eu1  2225  eqss  3288