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  3392  reuanidOLD  3393  axrep5  5287  elinintrab  43590  2sb5nd  44580  eelTT1  44730  uun121  44803  uunTT1  44813  uunTT1p1  44814  uunTT1p2  44815  uun111  44825  uun2221  44833  uun2221p1  44834  uun2221p2  44835  2sb5ndVD  44930  2sb5ndALT  44952