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  3484  elni2  7517  elpq  9861  elfz0ubfz0  10338  elfzmlbp  10345  fzo1fzo0n0  10400  elfzo0z  10401  fzofzim  10405  elfzodifsumelfzo  10424  swrdswrd  11258  swrdccatin1  11278  p1modz1  12326  dfgcd2  12556  algcvga  12594  pcprendvds  12834  usgruspgrben  16005  trlf1  16157