Theorem joinimandinc3-P4 578

Description: Alternate form of joinimandinc-P4.8a 397. †

Hypotheses

Ref	Expression
joinimandinc3-P4.1	⊢ (𝛾 → (𝜑 → 𝜓))
joinimandinc3-P4.2	⊢ (𝛾 → (𝜒 → 𝜓))

Assertion

Ref	Expression
joinimandinc3-P4	⊢ (𝛾 → ((𝜑 ∨ 𝜒) → 𝜓))

Proof of Theorem joinimandinc3-P4

Step	Hyp	Ref	Expression
1		joinimandinc3-P4.1	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜓))
2		joinimandinc3-P4.2	. . . 4 ⊢ (𝛾 → (𝜒 → 𝜓))
3	1, 2	ndandi-P3.7 172	. . 3 ⊢ (𝛾 → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
4	3	joinimandinc-P4.8a 397	. 2 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜓)))
5		idempotor-P4.25b 451	. . 3 ⊢ ((𝜓 ∨ 𝜓) ↔ 𝜓)
6	5	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ((𝜓 ∨ 𝜓) ↔ 𝜓))
7	4, 6	subimr2-P4 542	1 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) → 𝜓))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∨ wff-or 144

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  joinimandinc3-P4.RC  579