Theorem abai 839

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 363	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
2	1	pm5.32i 585	1 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  abab  840  pm5.75  1046  exintrbi  1924  dfeumo  2562  eu6  2600  dfeu  2621  r19.29imd  3128  dfss4  4215  indifdi  4240  choc0  31928  eu6w  43687