Theorem subandd-P3.42c 343

Description: Dual Substitution Law for '∧'. †

Hypotheses

Ref	Expression
subandd-P3.42c.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))
subandd-P3.42c.2	⊢ (𝛾 → (𝜒 ↔ 𝜗))

Assertion

Ref	Expression
subandd-P3.42c	⊢ (𝛾 → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜗)))

Proof of Theorem subandd-P3.42c

Step	Hyp	Ref	Expression
1		subandd-P3.42c.1	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	subandl-P3.42a 339	. 2 ⊢ (𝛾 → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)))
3		subandd-P3.42c.2	. . 3 ⊢ (𝛾 → (𝜒 ↔ 𝜗))
4	3	subandr-P3.42b 341	. 2 ⊢ (𝛾 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜗)))
5	2, 4	bitrns-P3.33c 302	1 ⊢ (𝛾 → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜗)))

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  df-true-D2.4 155

This theorem is referenced by:  subandd-P3.42c.RC  344  subandd2-P4  556