Theorem subbil-P3.41a-L1 331

Description: Lemma for subbil-P3.41a 332. †

Hypothesis

Ref	Expression
subbil-P3.41a-L1.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subbil-P3.41a-L1	⊢ (𝛾 → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))

Proof of Theorem subbil-P3.41a-L1

Step	Hyp	Ref	Expression
1		subbil-P3.41a-L1.1	. . . . 5 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	rcp-NDIMP1add1 208	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ↔ 𝜒)) → (𝜑 ↔ 𝜓))
3	2	bisym-P3.33b 298	. . 3 ⊢ ((𝛾 ∧ (𝜑 ↔ 𝜒)) → (𝜓 ↔ 𝜑))
4		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ (𝜑 ↔ 𝜒)) → (𝜑 ↔ 𝜒))
5	3, 4	bitrns-P3.33c 302	. 2 ⊢ ((𝛾 ∧ (𝜑 ↔ 𝜒)) → (𝜓 ↔ 𝜒))
6	5	rcp-NDIMI2 224	1 ⊢ (𝛾 → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∧ 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:  subbil-P3.41a  332