Theorem subiml-P3.40a 325

Description: Left Substitution Law for '→'. †

Hypothesis

Ref	Expression
subiml-P3.40a.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subiml-P3.40a	⊢ (𝛾 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))

Proof of Theorem subiml-P3.40a

Step	Hyp	Ref	Expression
1		subiml-P3.40a.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜓 → 𝜑))
3	2	imsubl-P3.28a 267	. 2 ⊢ (𝛾 → ((𝜑 → 𝜒) → (𝜓 → 𝜒)))
4	1	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
5	4	imsubl-P3.28a 267	. 2 ⊢ (𝛾 → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
6	3, 5	ndbii-P3.13 178	1 ⊢ (𝛾 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))

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:  subiml-P3.40a.RC  326  subiml2-P4  540  example-E5.04a  675  example-E6.01a  706  psubjust-P6  715  dfpsubv-P6  717