Theorem subandl-P3.42a 339

Description: Left Substitution Law for '∧'. †

Hypothesis

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

Assertion

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

Proof of Theorem subandl-P3.42a

Step	Hyp	Ref	Expression
1		subandl-P3.42a.1	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	subandl-P3.42a-L1 338	. 2 ⊢ (𝛾 → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
3	1	bisym-P3.33b 298	. . 3 ⊢ (𝛾 → (𝜓 ↔ 𝜑))
4	3	subandl-P3.42a-L1 338	. 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   ∧ 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:  subandl-P3.42a.RC  340  subandr-P3.42b  341  subandd-P3.42c  343  subandl2-P4  552