Theorem ifpnotnotb 44265

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 44264	. 2 ⊢ (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
2	1	bicomi 227	1 ⊢ (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  ifpananb  44292  ifpnannanb  44293  ifpxorxorb  44297