Theorem anabsi5 568

Description: Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)

Hypothesis

Ref	Expression
anabsi5.1	⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))

Assertion

Ref	Expression
anabsi5	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem anabsi5

Step	Hyp	Ref	Expression
1		anabsi5.1	. . 3 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))
2	1	imp 123	. 2 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒)
3	2	anabss5 567	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  anabsi6  569  anabsi8  571  3anidm12  1273  equsexd  1707  rspce  2779  phplem3g  6743  ltexprlemrl  7411  ltexprlemru  7413  dvdssq  11708