Theorem subiml-P2.8a 128

Description: Left Substitution Law for '→'.

Assertion

Ref	Expression
subiml-P2.8a	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))

Proof of Theorem subiml-P2.8a

Step	Hyp	Ref	Expression
1		birev-P2.5b 115	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2	1	imsubl-P1.7b.AC.SH 56	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) → (𝜓 → 𝜒)))
3		bifwd-P2.5a 111	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	imsubl-P1.7b.AC.SH 56	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
5	2, 4	bicmb-P2.5c.AC.2SH 121	1 ⊢ ((𝜑 ↔ 𝜓) → ((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

This theorem is referenced by:  subiml-P2.8a.SH  129