Theorem ifpbi1b

Description: When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020)

	Ref	Expression
Assertion	ifpbi1b	⊢ ( if- ( 𝜑 , 𝜒 , 𝜒 ) ↔ if- ( 𝜓 , 𝜒 , 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝜒 → 𝜒 )
2	1	olci	⊢ ( ( 𝜑 → ¬ 𝜓 ) ∨ ( 𝜒 → 𝜒 ) )
3	1	olci	⊢ ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜒 ) )
4	2 3	pm3.2i	⊢ ( ( ( 𝜑 → ¬ 𝜓 ) ∨ ( 𝜒 → 𝜒 ) ) ∧ ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜒 ) ) )
5	1	olci	⊢ ( ( 𝜑 → 𝜓 ) ∨ ( 𝜒 → 𝜒 ) )
6	1	olci	⊢ ( ( ¬ 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜒 ) )
7	5 6	pm3.2i	⊢ ( ( ( 𝜑 → 𝜓 ) ∨ ( 𝜒 → 𝜒 ) ) ∧ ( ( ¬ 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜒 ) ) )
8		ifpim123g	⊢ ( ( if- ( 𝜑 , 𝜒 , 𝜒 ) → if- ( 𝜓 , 𝜒 , 𝜒 ) ) ↔ ( ( ( ( 𝜑 → ¬ 𝜓 ) ∨ ( 𝜒 → 𝜒 ) ) ∧ ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜒 ) ) ) ∧ ( ( ( 𝜑 → 𝜓 ) ∨ ( 𝜒 → 𝜒 ) ) ∧ ( ( ¬ 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜒 ) ) ) ) )
9	4 7 8	mpbir2an	⊢ ( if- ( 𝜑 , 𝜒 , 𝜒 ) → if- ( 𝜓 , 𝜒 , 𝜒 ) )
10	1	olci	⊢ ( ( 𝜓 → ¬ 𝜑 ) ∨ ( 𝜒 → 𝜒 ) )
11	10 5	pm3.2i	⊢ ( ( ( 𝜓 → ¬ 𝜑 ) ∨ ( 𝜒 → 𝜒 ) ) ∧ ( ( 𝜑 → 𝜓 ) ∨ ( 𝜒 → 𝜒 ) ) )
12	1	olci	⊢ ( ( ¬ 𝜑 → 𝜓 ) ∨ ( 𝜒 → 𝜒 ) )
13	3 12	pm3.2i	⊢ ( ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜒 ) ) ∧ ( ( ¬ 𝜑 → 𝜓 ) ∨ ( 𝜒 → 𝜒 ) ) )
14		ifpim123g	⊢ ( ( if- ( 𝜓 , 𝜒 , 𝜒 ) → if- ( 𝜑 , 𝜒 , 𝜒 ) ) ↔ ( ( ( ( 𝜓 → ¬ 𝜑 ) ∨ ( 𝜒 → 𝜒 ) ) ∧ ( ( 𝜑 → 𝜓 ) ∨ ( 𝜒 → 𝜒 ) ) ) ∧ ( ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜒 ) ) ∧ ( ( ¬ 𝜑 → 𝜓 ) ∨ ( 𝜒 → 𝜒 ) ) ) ) )
15	11 13 14	mpbir2an	⊢ ( if- ( 𝜓 , 𝜒 , 𝜒 ) → if- ( 𝜑 , 𝜒 , 𝜒 ) )
16	9 15	impbii	⊢ ( if- ( 𝜑 , 𝜒 , 𝜒 ) ↔ if- ( 𝜓 , 𝜒 , 𝜒 ) )