Theorem subimd2-P4.RC 545

Description: Inference Form of subimd2-P4 544. †

Hypotheses

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

Assertion

Ref	Expression
subimd2-P4.RC	⊢ (𝜓 → 𝜗)

Proof of Theorem subimd2-P4.RC

Step	Hyp	Ref	Expression
1		subimd2-P4.RC.1	. . . 4 ⊢ (𝜑 → 𝜒)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 → 𝜒))
3		subimd2-P4.RC.2	. . . 4 ⊢ (𝜑 ↔ 𝜓)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
5		subimd2-P4.RC.3	. . . 4 ⊢ (𝜒 ↔ 𝜗)
6	5	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜒 ↔ 𝜗))
7	2, 4, 6	subimd2-P4 544	. 2 ⊢ (⊤ → (𝜓 → 𝜗))
8	7	ndtruee-P3.18 183	1 ⊢ (𝜓 → 𝜗)

Syntax hints:   → wff-imp 10   ↔ wff-bi 104  ⊤wff-true 153

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-true-D2.4 155

This theorem is referenced by:  lemma-L5.05a  668  genallw-P6  676  genexw-P6  677  gennallw-P6  678  gennexw-P6  679  gennex-P6  738  nfrexgencl-L6  813