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  954  pm5.63dc  955  simplbi2com  1490  reuss2  3500  elni2  7628  elpq  9980  elfz0ubfz0  10458  elfzmlbp  10465  fzo1fzo0n0  10521  elfzo0z  10522  fzofzim  10526  elfzodifsumelfzo  10545  swrdswrd  11393  swrdccatin1  11413  p1modz1  12476  dfgcd2  12706  algcvga  12744  pcprendvds  12984  usgruspgrben  16173  trlf1  16375