Theorem simplbi2com 1494

Description: A deduction eliminating a conjunct, similar to simplbi2 385. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)

Hypothesis

Ref	Expression
simplbi2com.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
simplbi2com	⊢ (𝜒 → (𝜓 → 𝜑))

Proof of Theorem simplbi2com

Step	Hyp	Ref	Expression
1		simplbi2com.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2	1	simplbi2 385	. 2 ⊢ (𝜓 → (𝜒 → 𝜑))
3	2	com12 30	1 ⊢ (𝜒 → (𝜓 → 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  mo2r  2139  mo3h  2140  elres  5099  xpidtr  5178  elovmporab  6289  elovmporab1w  6290  peano5nnnn  8259  peano5nni  9308  modprmn0modprm0  13037  insubm  13794  cnptoprest  15342