Theorem subbil-P3.41a.RC 333

Description: Inference Form of subbil-P3.41a 332. †

Hypothesis

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

Assertion

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

Proof of Theorem subbil-P3.41a.RC

Step	Hyp	Ref	Expression
1		subbil-P3.41a.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3	2	subbil-P3.41a 332	. 2 ⊢ (⊤ → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))
4	3	ndtruee-P3.18 183	1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))

Syntax hints:   ↔ wff-bi 104  ⊤wff-true 153

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  df-true-D2.4 155

This theorem is referenced by:  idandtruer-P4.19b  439  idorfalser-P4.20b  441  idandthml-P4.23a  446  idandthmr-P4.23b  447  idornthml-P4.24a  448  idornthmr-P4.24b  449