Theorem joinimandres3-P4 582

Description: Alternate form of joinimandres-P4.8b 400. †

Hypotheses

Ref	Expression
joinimandres3-P4.1	⊢ (𝛾 → (𝜑 → 𝜓))
joinimandres3-P4.2	⊢ (𝛾 → (𝜑 → 𝜒))

Assertion

Ref	Expression
joinimandres3-P4	⊢ (𝛾 → (𝜑 → (𝜓 ∧ 𝜒)))

Proof of Theorem joinimandres3-P4

Step	Hyp	Ref	Expression
1		joinimandres3-P4.1	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜓))
2		joinimandres3-P4.2	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜒))
3	1, 2	ndandi-P3.7 172	. . 3 ⊢ (𝛾 → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))
4	3	joinimandres-P4.8b 400	. 2 ⊢ (𝛾 → ((𝜑 ∧ 𝜑) → (𝜓 ∧ 𝜒)))
5		idempotand-P4.25a 450	. . 3 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
6	5	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ((𝜑 ∧ 𝜑) ↔ 𝜑))
7	4, 6	subiml2-P4 540	1 ⊢ (𝛾 → (𝜑 → (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  df-true-D2.4 155

This theorem is referenced by:  joinimandres3-P4.RC  583