Theorem abai 832

Description: Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)

Assertion

Ref	Expression
abai	⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓)))

Proof of Theorem abai

Step	Hyp	Ref	Expression
1		biimt 361	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2	1	pm5.32i 579	1 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396

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

This theorem depends on definitions:  df-bi 208  df-an 397

This theorem is referenced by:  pm5.75  1036  exintrbi  1898  dfeumo  2540  eu6  2578  dfeu  2599  r19.29imd  3104  dfss4  4197  indifdi  4222  choc0  31415  eu6w  43126