Theorem abai 770

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 325	. 2 ⊢ (φ → (ψ ↔ (φ → ψ)))
2	1	pm5.32i 618	1 ⊢ ((φ ∧ ψ) ↔ (φ ∧ (φ → ψ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358

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

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

This theorem is used by:  eu2  2229  2eu6  2289  dfss4  3490