Theorem ifpbi23 42209

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)

Assertion

Ref	Expression
ifpbi23	⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))

Proof of Theorem ifpbi23

Step	Hyp	Ref	Expression
1		simpl 483	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓))
2		simpr 485	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃))
3	1, 2	ifpbi23d 1080	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  if-wif 1061

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 206  df-an 397  df-or 846  df-ifp 1062

This theorem is referenced by:  ifpnot23d  42221  ifpdfxor  42223  ifpananb  42242  ifpnannanb  42243  ifpxorxorb  42247