Theorem ifpbi123d 1021

Description: Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.)

Hypotheses

Ref	Expression
ifpbi123d.1	⊢ (𝜑 → (𝜓 ↔ 𝜏))
ifpbi123d.2	⊢ (𝜑 → (𝜒 ↔ 𝜂))
ifpbi123d.3	⊢ (𝜑 → (𝜃 ↔ 𝜁))

Assertion

Ref	Expression
ifpbi123d	⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Proof of Theorem ifpbi123d

Step	Hyp	Ref	Expression
1		ifpbi123d.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜏))
2		ifpbi123d.2	. . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜂))
3	1, 2	anbi12d 743	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜏 ∧ 𝜂)))
4	1	notbid 307	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜏))
5		ifpbi123d.3	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜁))
6	4, 5	anbi12d 743	. . 3 ⊢ (𝜑 → ((¬ 𝜓 ∧ 𝜃) ↔ (¬ 𝜏 ∧ 𝜁)))
7	3, 6	orbi12d 742	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜃)) ↔ ((𝜏 ∧ 𝜂) ∨ (¬ 𝜏 ∧ 𝜁))))
8		df-ifp 1007	. 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜃)))
9		df-ifp 1007	. 2 ⊢ (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏 ∧ 𝜂) ∨ (¬ 𝜏 ∧ 𝜁)))
10	7, 8, 9	3bitr4g 302	1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  if-wif 1006

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 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by:  1wlkslem1  40809  1wlkslem2  40810  1wlksfval  40811  is1wlk  40813  1wlkres  40879  red1wlk  40881  1wlkp1lem8  40889  crctcsh1wlkn0lem4  41016  crctcsh1wlkn0lem5  41017  crctcsh1wlkn0lem6  41018  11wlkdlem4  41307