Theorem subimd-P3.40c 329

Description: Dual Substitution Law for '→'. †

Hypotheses

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

Assertion

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

Proof of Theorem subimd-P3.40c

Step	Hyp	Ref	Expression
1		subimd-P3.40c.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜓 → 𝜑))
3		subimd-P3.40c.2	. . . 4 ⊢ (𝛾 → (𝜒 ↔ 𝜗))
4	3	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜒 → 𝜗))
5	2, 4	imsubd-P3.28c 271	. 2 ⊢ (𝛾 → ((𝜑 → 𝜒) → (𝜓 → 𝜗)))
6	1	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
7	3	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜗 → 𝜒))
8	6, 7	imsubd-P3.28c 271	. 2 ⊢ (𝛾 → ((𝜓 → 𝜗) → (𝜑 → 𝜒)))
9	5, 8	ndbii-P3.13 178	1 ⊢ (𝛾 → ((𝜑 → 𝜒) ↔ (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:  subimd-P3.40c.RC  330  subimd2-P4  544  example-E5.03a  665  example-E5.04a  675  example-E6.01a  706  psubjust-P6  715  subnfr-P6  755