Theorem subimr-P3.40b 327

Description: Right Substitution Theorem for '→'. †

Hypothesis

Ref	Expression
subimr-P3.40b.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subimr-P3.40b	⊢ (𝛾 → ((𝜒 → 𝜑) ↔ (𝜒 → 𝜓)))

Proof of Theorem subimr-P3.40b

Step	Hyp	Ref	Expression
1		subimr-P3.40b.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
3	2	imsubr-P3.28b 269	. 2 ⊢ (𝛾 → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))
4	1	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜓 → 𝜑))
5	4	imsubr-P3.28b 269	. 2 ⊢ (𝛾 → ((𝜒 → 𝜓) → (𝜒 → 𝜑)))
6	3, 5	ndbii-P3.13 178	1 ⊢ (𝛾 → ((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  df-and-D2.2 133

This theorem is referenced by:  subimr-P3.40b.RC  328  subimr2-P4  542