Theorem subimd-P3.40c.RC 330

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

Hypotheses

Ref	Expression
subimd-P3.40c.RC.1	⊢ (𝜑 ↔ 𝜓)
subimd-P3.40c.RC.2	⊢ (𝜒 ↔ 𝜗)

Assertion

Ref	Expression
subimd-P3.40c.RC	⊢ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜗))

Proof of Theorem subimd-P3.40c.RC

Step	Hyp	Ref	Expression
1		subimd-P3.40c.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3		subimd-P3.40c.RC.2	. . . 4 ⊢ (𝜒 ↔ 𝜗)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜒 ↔ 𝜗))
5	2, 4	subimd-P3.40c 329	. 2 ⊢ (⊤ → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜗)))
6	5	ndtruee-P3.18 183	1 ⊢ ((𝜑 → 𝜒) ↔ (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:  subeqr-P5-L1  634  solvesub-P6a  704  lemma-L6.02a  726  genex-P6  731  psubnfr-P6  784  psubim-P6-L1  789