Theorem ifpnotnotb 39838

Description: Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)

Assertion

Ref	Expression
ifpnotnotb	⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒))

Proof of Theorem ifpnotnotb

Step	Hyp	Ref	Expression
1		ifpnot23 39837	. 2 ⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
2	1	bicomi 226	1 ⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 208  if-wif 1057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058

This theorem is referenced by:  ifpananb  39865  ifpnannanb  39866  ifpxorxorb  39870