Theorem subandr-P3.42b.RC 342

Description: Inference Form of subandr-P3.42b 341. †

Hypothesis

Ref	Expression
subandr-P3.42b.RC.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
subandr-P3.42b.RC	⊢ ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓))

Proof of Theorem subandr-P3.42b.RC

Step	Hyp	Ref	Expression
1		subandr-P3.42b.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3	2	subandr-P3.42b 341	. 2 ⊢ (⊤ → ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓)))
4	3	ndtruee-P3.18 183	1 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓))

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:  idandthmr-P4.23b  447  lemma-L7.02a-L1  943