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Theorem ifpbi23 37643
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
Assertion
Ref Expression
ifpbi23 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))

Proof of Theorem ifpbi23
StepHypRef Expression
1 simpl 473 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜓))
21imbi2d 330 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜏𝜑) ↔ (𝜏𝜓)))
3 simpr 477 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜒𝜃))
43imbi2d 330 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((¬ 𝜏𝜒) ↔ (¬ 𝜏𝜃)))
52, 4anbi12d 747 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (((𝜏𝜑) ∧ (¬ 𝜏𝜒)) ↔ ((𝜏𝜓) ∧ (¬ 𝜏𝜃))))
6 dfifp2 1014 . 2 (if-(𝜏, 𝜑, 𝜒) ↔ ((𝜏𝜑) ∧ (¬ 𝜏𝜒)))
7 dfifp2 1014 . 2 (if-(𝜏, 𝜓, 𝜃) ↔ ((𝜏𝜓) ∧ (¬ 𝜏𝜃)))
85, 6, 73bitr4g 303 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  if-wif 1012
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 197  df-or 385  df-an 386  df-ifp 1013
This theorem is referenced by:  ifpdfbi  37644  ifpnot23d  37656  ifpdfxor  37658  ifpananb  37677  ifpnannanb  37678  ifpxorxorb  37682
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