Theorem subord2-P4 562

Description: Alternate Form of subord-P3.43c 350. †

Hypotheses

Ref	Expression
subord2-P4.1	⊢ (𝛾 → (𝜑 ∨ 𝜒))
subord2-P4.2	⊢ (𝛾 → (𝜑 ↔ 𝜓))
subord2-P4.3	⊢ (𝛾 → (𝜒 ↔ 𝜗))

Assertion

Ref	Expression
subord2-P4	⊢ (𝛾 → (𝜓 ∨ 𝜗))

Proof of Theorem subord2-P4

Step	Hyp	Ref	Expression
1		subord2-P4.1	. 2 ⊢ (𝛾 → (𝜑 ∨ 𝜒))
2		subord2-P4.2	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3		subord2-P4.3	. . . 4 ⊢ (𝛾 → (𝜒 ↔ 𝜗))
4	2, 3	subord-P3.43c 350	. . 3 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜗)))
5	4	ndbief-P3.14 179	. 2 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜗)))
6	1, 5	ndime-P3.6 171	1 ⊢ (𝛾 → (𝜓 ∨ 𝜗))

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:  subord2-P4.RC  563