Theorem imcomm-P1.6 48

Description: Commutation of Antecedents.

Assertion

Ref	Expression
imcomm-P1.6	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Proof of Theorem imcomm-P1.6

Step	Hyp	Ref	Expression
1		ax-L2 12	. . . 4 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2	1	axL1.AC.SH 45	. . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
3	2	axL2.AC.SH 46	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜓 → (𝜑 → 𝜓)) → (𝜓 → (𝜑 → 𝜒))))
4		ax-L1 11	. 2 ⊢ (𝜓 → (𝜑 → 𝜓))
5	3, 4	mae-P1.1 33	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Syntax hints:   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-MP 14

This theorem is referenced by:  imcomm-P1.6.SH  49  imcomm-P1.6.AC.SH  50  maet-P1.10  64