Theorem subandd2-P4 556

Description: Alternate Form of subandd-P3.42c 343. †

Hypotheses

Ref	Expression
subandd2-P4.1	⊢ (𝛾 → (𝜑 ∧ 𝜒))
subandd2-P4.2	⊢ (𝛾 → (𝜑 ↔ 𝜓))
subandd2-P4.3	⊢ (𝛾 → (𝜒 ↔ 𝜗))

Assertion

Ref	Expression
subandd2-P4	⊢ (𝛾 → (𝜓 ∧ 𝜗))

Proof of Theorem subandd2-P4

Step	Hyp	Ref	Expression
1		subandd2-P4.1	. 2 ⊢ (𝛾 → (𝜑 ∧ 𝜒))
2		subandd2-P4.2	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3		subandd2-P4.3	. . . 4 ⊢ (𝛾 → (𝜒 ↔ 𝜗))
4	2, 3	subandd-P3.42c 343	. . 3 ⊢ (𝛾 → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜗)))
5	4	ndbief-P3.14 179	. 2 ⊢ (𝛾 → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜗)))
6	1, 5	ndime-P3.6 171	1 ⊢ (𝛾 → (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:  subandd2-P4.RC  557