Theorem albiim 1887

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 474	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
2	1	albii 1816	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ ∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
3		19.26 1868	. 2 ⊢ (∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))
4	2, 3	bitri 275	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  2albiim  1888  dfmoeu  2534  mobi  2545  eu6lem  2571  eu1  2608  eqss  4011  rabeqsnd  4674  ssext  5465  asymref2  6140  eu6w  42663  pm14.122a  44418