Theorem subandl-P3.42a.RC 340

Description: Inference Form of subandl-P3.42a 339. †

Hypothesis

Ref	Expression
subandl-P3.42a.RC.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
subandl-P3.42a.RC	⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))

Proof of Theorem subandl-P3.42a.RC

Step	Hyp	Ref	Expression
1		subandl-P3.42a.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3	2	subandl-P3.42a 339	. 2 ⊢ (⊤ → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)))
4	3	ndtruee-P3.18 183	1 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))

Syntax hints:   ↔ wff-bi 104   ∧ wff-and 132  ⊤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:  idandthml-P4.23a  446