Theorem simplbi2 385

Description: Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)

Hypothesis

Ref	Expression
pm3.26bi2.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
simplbi2	⊢ (𝜓 → (𝜒 → 𝜑))

Proof of Theorem simplbi2

Step	Hyp	Ref	Expression
1		pm3.26bi2.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2	1	biimpri 133	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
3	2	ex 115	1 ⊢ (𝜓 → (𝜒 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.62dc  951  pm5.63dc  952  simplbi2com  1487  reuss2  3485  elni2  7527  elpq  9876  elfz0ubfz0  10353  elfzmlbp  10360  fzo1fzo0n0  10415  elfzo0z  10416  fzofzim  10420  elfzodifsumelfzo  10439  swrdswrd  11279  swrdccatin1  11299  p1modz1  12348  dfgcd2  12578  algcvga  12616  pcprendvds  12856  usgruspgrben  16030  trlf1  16197