Theorem cmb-P2.9c 138

Description: '∧' Introduction by Combination.

Assertion

Ref	Expression
cmb-P2.9c	⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))

Proof of Theorem cmb-P2.9c

Step	Hyp	Ref	Expression
1		cmb-L2.3 99	. 2 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
2		df-and-D2.2 133	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
3	2	subimr-P2.8b.SH 131	. . . 4 ⊢ ((𝜓 → (𝜑 ∧ 𝜓)) ↔ (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
4	3	subimr-P2.8b.SH 131	. . 3 ⊢ ((𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) ↔ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓))))
5	4	birev-P2.5b.SH 116	. 2 ⊢ ((𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓))) → (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))))
6	1, 5	ax-MP 14	1 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wff-neg 9   → 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:  cmb-P2.9c.AC.2SH  139  export-P2.10b  142