Theorem subimr-P2.8b 130

Description: Right Substitution Law for '→'.

Assertion

Ref	Expression
subimr-P2.8b	⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓)))

Proof of Theorem subimr-P2.8b

Step	Hyp	Ref	Expression
1		bifwd-P2.5a 111	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	imsubr-P1.7a.AC.SH 53	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))
3		birev-P2.5b 115	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
4	3	imsubr-P1.7a.AC.SH 53	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜓) → (𝜒 → 𝜑)))
5	2, 4	bicmb-P2.5c.AC.2SH 121	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓)))

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:  subimr-P2.8b.SH  131