Theorem ifpimim 41116

Description: Consequnce of implication. (Contributed by RP, 17-Apr-2020.)

Assertion

Ref	Expression
ifpimim	⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))

Proof of Theorem ifpimim

Step	Hyp	Ref	Expression
1		pm2.521 176	. . . . . 6 ⊢ (¬ (¬ 𝜑 → 𝜑) → (𝜑 → ¬ 𝜑))
2	1	orim1i 907	. . . . 5 ⊢ ((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) → ((𝜑 → ¬ 𝜑) ∨ (𝜓 → 𝜒)))
3	2	adantr 481	. . . 4 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → ((𝜑 → ¬ 𝜑) ∨ (𝜓 → 𝜒)))
4		id 22	. . . . . 6 ⊢ (𝜑 → 𝜑)
5	4	orci 862	. . . . 5 ⊢ ((𝜑 → 𝜑) ∨ (𝜃 → 𝜒))
6	5	a1i 11	. . . 4 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → ((𝜑 → 𝜑) ∨ (𝜃 → 𝜒)))
7	3, 6	jca 512	. . 3 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → (((𝜑 → ¬ 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((𝜑 → 𝜑) ∨ (𝜃 → 𝜒))))
8	4	orci 862	. . . . 5 ⊢ ((𝜑 → 𝜑) ∨ (𝜓 → 𝜏))
9	8	a1i 11	. . . 4 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → ((𝜑 → 𝜑) ∨ (𝜓 → 𝜏)))
10		simpr 485	. . . 4 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏)))
11	9, 10	jca 512	. . 3 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → (((𝜑 → 𝜑) ∨ (𝜓 → 𝜏)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))))
12	7, 11	jca 512	. 2 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → ((((𝜑 → ¬ 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((𝜑 → 𝜑) ∨ (𝜃 → 𝜒))) ∧ (((𝜑 → 𝜑) ∨ (𝜓 → 𝜏)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏)))))
13		pm4.81 394	. . . . 5 ⊢ ((¬ 𝜑 → 𝜑) ↔ 𝜑)
14	13	bicomi 223	. . . 4 ⊢ (𝜑 ↔ (¬ 𝜑 → 𝜑))
15		ifpbi1 41084	. . . 4 ⊢ ((𝜑 ↔ (¬ 𝜑 → 𝜑)) → (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ if-((¬ 𝜑 → 𝜑), (𝜓 → 𝜒), (𝜃 → 𝜏))))
16	14, 15	ax-mp 5	. . 3 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ if-((¬ 𝜑 → 𝜑), (𝜓 → 𝜒), (𝜃 → 𝜏)))
17		dfifp4 1064	. . 3 ⊢ (if-((¬ 𝜑 → 𝜑), (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ ((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))))
18	16, 17	bitri 274	. 2 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ ((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))))
19		ifpim123g 41107	. 2 ⊢ ((if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)) ↔ ((((𝜑 → ¬ 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((𝜑 → 𝜑) ∨ (𝜃 → 𝜒))) ∧ (((𝜑 → 𝜑) ∨ (𝜓 → 𝜏)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏)))))
20	12, 18, 19	3imtr4i 292	1 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  if-wif 1060

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 845  df-ifp 1061

This theorem is referenced by:  frege58acor  41484  frege60a  41486  frege65a  41491