Theorem re1ax2lem 33730

Description: Lemma for re1ax2 33731. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1ax2lem	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Proof of Theorem re1ax2lem

Step	Hyp	Ref	Expression
1		tb-ax2 33727	. . . 4 ⊢ (𝜓 → ((𝜓 → 𝜒) → 𝜓))
2		tb-ax1 33726	. . . 4 ⊢ (((𝜓 → 𝜒) → 𝜓) → ((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
3	1, 2	tbsyl 33729	. . 3 ⊢ (𝜓 → ((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
4		tb-ax1 33726	. . . 4 ⊢ (((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((((𝜓 → 𝜒) → 𝜒) → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
5		tb-ax3 33728	. . . 4 ⊢ (((((𝜓 → 𝜒) → 𝜒) → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((𝜓 → 𝜒) → 𝜒))
6	4, 5	tbsyl 33729	. . 3 ⊢ (((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((𝜓 → 𝜒) → 𝜒))
7	3, 6	tbsyl 33729	. 2 ⊢ (𝜓 → ((𝜓 → 𝜒) → 𝜒))
8		tb-ax1 33726	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)))
9		tb-ax1 33726	. 2 ⊢ ((𝜓 → ((𝜓 → 𝜒) → 𝜒)) → ((((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))))
10	7, 8, 9	mpsyl 68	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  re1ax2  33731