Theorem anabs5 663

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 528	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
2	1	bicomd 223	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓))
3	2	pm5.32i 574	1 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  rmoanidOLD  3357  reuanidOLD  3358  axrep5  5229  elinintrab  43553  2sb5nd  44537  eelTT1  44686  uun121  44759  uunTT1  44769  uunTT1p1  44770  uunTT1p2  44771  uun111  44781  uun2221  44789  uun2221p1  44790  uun2221p2  44791  2sb5ndVD  44886  2sb5ndALT  44908