Theorem imsubl-P3.28a 267

Description: Implication Substitution (left). †

Hypothesis

Ref	Expression
imsubl-P3.28a.1	⊢ (𝛾 → (𝜑 → 𝜓))

Assertion

Ref	Expression
imsubl-P3.28a	⊢ (𝛾 → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Proof of Theorem imsubl-P3.28a

Step	Hyp	Ref	Expression
1		imsubl-P3.28a.1	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜓))
2	1	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ (𝜓 → 𝜒)) → (𝜑 → 𝜓))
3		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ (𝜓 → 𝜒)) → (𝜓 → 𝜒))
4	2, 3	syl-P3.24 259	. 2 ⊢ ((𝛾 ∧ (𝜓 → 𝜒)) → (𝜑 → 𝜒))
5	4	rcp-NDIMI2 224	1 ⊢ (𝛾 → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132

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:  imsubl-P3.28a.RC  268  imsubd-P3.28c  271  subiml-P3.40a  325