Theorem anabs5 675

Description: Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)

Assertion

Ref	Expression
anabs5	⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))

Proof of Theorem anabs5

Step	Hyp	Ref	Expression
1		ibar 537	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
2	1	bicomd 226	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓))
3	2	pm5.32i 584	1 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 400

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 210  df-an 401

This theorem is referenced by:  axrep5  5246  elinintrab  44303  2sb5nd  45269  eelTT1  45418  uun121  45491  uunTT1  45501  uunTT1p1  45502  uunTT1p2  45503  uun111  45513  uun2221  45521  uun2221p1  45522  uun2221p2  45523  2sb5ndVD  45618  2sb5ndALT  45640