Theorem simplbi2 377

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 131	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
3	2	ex 113	1 ⊢ (𝜓 → (𝜒 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm5.62dc  886  pm5.63dc  887  simplbi2com  1373  reuss2  3244  elni2  6504  elfz0ubfz0  9136  elfzmlbp  9143  fzo1fzo0n0  9192  elfzo0z  9193  fzofzim  9197  elfzodifsumelfzo  9210  dfgcd2  10403  ialgcvga  10433