Theorem ifpbi23 37817

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 473	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓))
2	1	imbi2d 330	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
3		simpr 477	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃))
4	3	imbi2d 330	. . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((¬ 𝜏 → 𝜒) ↔ (¬ 𝜏 → 𝜃)))
5	2, 4	anbi12d 747	. 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (((𝜏 → 𝜑) ∧ (¬ 𝜏 → 𝜒)) ↔ ((𝜏 → 𝜓) ∧ (¬ 𝜏 → 𝜃))))
6		dfifp2 1014	. 2 ⊢ (if-(𝜏, 𝜑, 𝜒) ↔ ((𝜏 → 𝜑) ∧ (¬ 𝜏 → 𝜒)))
7		dfifp2 1014	. 2 ⊢ (if-(𝜏, 𝜓, 𝜃) ↔ ((𝜏 → 𝜓) ∧ (¬ 𝜏 → 𝜃)))
8	5, 6, 7	3bitr4g 303	1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))

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  37818  ifpnot23d  37830  ifpdfxor  37832  ifpananb  37851  ifpnannanb  37852  ifpxorxorb  37856