Theorem cmb-P2.9c.AC.2SH 139

Description: Deductive Form of cmb-P2.9c 138.

Hypotheses

Ref	Expression
cmb-P2.9c.AC.2SH.1	⊢ (𝛾 → 𝜑)
cmb-P2.9c.AC.2SH.2	⊢ (𝛾 → 𝜓)

Assertion

Ref	Expression
cmb-P2.9c.AC.2SH	⊢ (𝛾 → (𝜑 ∧ 𝜓))

Proof of Theorem cmb-P2.9c.AC.2SH

Step	Hyp	Ref	Expression
1		cmb-P2.9c.AC.2SH.2	. 2 ⊢ (𝛾 → 𝜓)
2		cmb-P2.9c.AC.2SH.1	. . . 4 ⊢ (𝛾 → 𝜑)
3		cmb-P2.9c 138	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
4	3	axL1.SH 30	. . . . 5 ⊢ (𝛾 → (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))))
5	4	rcp-FR1.SH 40	. . . 4 ⊢ ((𝛾 → 𝜑) → (𝛾 → (𝜓 → (𝜑 ∧ 𝜓))))
6	2, 5	ax-MP 14	. . 3 ⊢ (𝛾 → (𝜓 → (𝜑 ∧ 𝜓)))
7	6	rcp-FR1.SH 40	. 2 ⊢ ((𝛾 → 𝜓) → (𝛾 → (𝜑 ∧ 𝜓)))
8	1, 7	ax-MP 14	1 ⊢ (𝛾 → (𝜑 ∧ 𝜓))

Syntax hints:   → wff-imp 10   ∧ 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:  ndandi-P3.7  172