Theorem ifpimim 44468

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		pm2.521 177	. . . . . 6 ⊢ (¬ (¬ 𝜑 → 𝜑) → (𝜑 → ¬ 𝜑))
2	1	orim1i 923	. . . . 5 ⊢ ((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) → ((𝜑 → ¬ 𝜑) ∨ (𝜓 → 𝜒)))
3	2	adantr 486	. . . 4 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → ((𝜑 → ¬ 𝜑) ∨ (𝜓 → 𝜒)))
4		id 23	. . . . . 6 ⊢ (𝜑 → 𝜑)
5	4	orci 879	. . . . 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 521	. . 3 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → (((𝜑 → ¬ 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((𝜑 → 𝜑) ∨ (𝜃 → 𝜒))))
8	4	orci 879	. . . . 5 ⊢ ((𝜑 → 𝜑) ∨ (𝜓 → 𝜏))
9	8	a1i 11	. . . 4 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → ((𝜑 → 𝜑) ∨ (𝜓 → 𝜏)))
10		simpr 490	. . . 4 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏)))
11	9, 10	jca 521	. . 3 ⊢ (((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))) → (((𝜑 → 𝜑) ∨ (𝜓 → 𝜏)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))))
12	7, 11	jca 521	. 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		pm4.81 399	. . . . 5 ⊢ ((¬ 𝜑 → 𝜑) ↔ 𝜑)
14	13	bicomi 227	. . . 4 ⊢ (𝜑 ↔ (¬ 𝜑 → 𝜑))
15		ifpbi1 44436	. . . 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 1082	. . 3 ⊢ (if-((¬ 𝜑 → 𝜑), (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ ((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))))
18	16, 17	bitri 278	. 2 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) ↔ ((¬ (¬ 𝜑 → 𝜑) ∨ (𝜓 → 𝜒)) ∧ ((¬ 𝜑 → 𝜑) ∨ (𝜃 → 𝜏))))
19		ifpim123g 44459	. 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 295	1 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079

This theorem is used by:  frege58acor  44835  frege60a  44837  frege65a  44842