Theorem anabsi5 513

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 115	. 2 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒)
3	2	anabss5 512	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 97

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  anabsi6  514  anabsi8  516  3anidm12  1192  equsexd  1617  rspce  2651  phplem3g  6319  ltexprlemrl  6708  ltexprlemru  6710