Theorem subbil-P3.41a 332

Description: Left Substitution Law for '↔'. †

Hypothesis

Ref	Expression
subbil-P3.41a.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subbil-P3.41a	⊢ (𝛾 → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))

Proof of Theorem subbil-P3.41a

Step	Hyp	Ref	Expression
1		subbil-P3.41a.1	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	subbil-P3.41a-L1 331	. 2 ⊢ (𝛾 → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))
3	1	bisym-P3.33b 298	. . 3 ⊢ (𝛾 → (𝜓 ↔ 𝜑))
4	3	subbil-P3.41a-L1 331	. 2 ⊢ (𝛾 → ((𝜓 ↔ 𝜒) → (𝜑 ↔ 𝜒)))
5	2, 4	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:  subbil-P3.41a.RC  333  subbir-P3.41b  334  subbid-P3.41c  336  subbil2-P4  546