Theorem albiim 1498

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 388	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
2	1	albii 1481	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ ∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
3		19.26 1492	. 2 ⊢ (∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))
4	2, 3	bitri 184	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  2albiim  1499  hbbid  1586  equveli  1770  spsbbi  1855  eu1  2067  eqss  3194  ssext  4250