Theorem anabs5 662

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 532	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
2	1	bicomd 226	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓))
3	2	pm5.32i 578	1 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 399

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 400

This theorem is referenced by:  reuanid  3286  rmoanid  3287  axrep5  5160  elinintrab  40277  2sb5nd  41266  eelTT1  41416  uun121  41489  uunTT1  41499  uunTT1p1  41500  uunTT1p2  41501  uun111  41511  uun2221  41519  uun2221p1  41520  uun2221p2  41521  2sb5ndVD  41616  2sb5ndALT  41638