Theorem subimd2-P4 544

Description: Alternate Form of subimd-P3.40c 329. †

Hypotheses

Ref	Expression
subimd2-P4.1	⊢ (𝛾 → (𝜑 → 𝜒))
subimd2-P4.2	⊢ (𝛾 → (𝜑 ↔ 𝜓))
subimd2-P4.3	⊢ (𝛾 → (𝜒 ↔ 𝜗))

Assertion

Ref	Expression
subimd2-P4	⊢ (𝛾 → (𝜓 → 𝜗))

Proof of Theorem subimd2-P4

Step	Hyp	Ref	Expression
1		subimd2-P4.1	. 2 ⊢ (𝛾 → (𝜑 → 𝜒))
2		subimd2-P4.2	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3		subimd2-P4.3	. . . 4 ⊢ (𝛾 → (𝜒 ↔ 𝜗))
4	2, 3	subimd-P3.40c 329	. . 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

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

This theorem is referenced by:  subimd2-P4.RC  545  alloverimex-P5  601