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  3505  elni2  7645  elpq  9999  elfz0ubfz0  10481  elfzmlbp  10488  fzo1fzo0n0  10544  elfzo0z  10545  fzofzim  10549  elfzodifsumelfzo  10568  swrdswrd  11422  swrdccatin1  11442  p1modz1  12505  dfgcd2  12735  algcvga  12773  pcprendvds  13013  usgruspgrben  16293  trlf1  16495