Theorem re1ax2 34504

Description: ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 34499 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1ax2	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Proof of Theorem re1ax2

Step	Hyp	Ref	Expression
1		re1ax2lem 34503	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
2		tb-ax1 34499	. . . 4 ⊢ ((𝜑 → (𝜑 → 𝜒)) → (((𝜑 → 𝜒) → 𝜒) → (𝜑 → 𝜒)))
3		tb-ax3 34501	. . . 4 ⊢ ((((𝜑 → 𝜒) → 𝜒) → (𝜑 → 𝜒)) → (𝜑 → 𝜒))
4	2, 3	tbsyl 34502	. . 3 ⊢ ((𝜑 → (𝜑 → 𝜒)) → (𝜑 → 𝜒))
5		tb-ax1 34499	. . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜑 → 𝜒))))
6		re1ax2lem 34503	. . . 4 ⊢ (((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜑 → 𝜒)))) → ((𝜓 → (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → (𝜑 → 𝜒)))))
7	5, 6	ax-mp 5	. . 3 ⊢ ((𝜓 → (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → (𝜑 → 𝜒))))
8		tb-ax1 34499	. . . 4 ⊢ (((𝜑 → 𝜓) → (𝜑 → (𝜑 → 𝜒))) → (((𝜑 → (𝜑 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
9		re1ax2lem 34503	. . . 4 ⊢ ((((𝜑 → 𝜓) → (𝜑 → (𝜑 → 𝜒))) → (((𝜑 → (𝜑 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) → (((𝜑 → (𝜑 → 𝜒)) → (𝜑 → 𝜒)) → (((𝜑 → 𝜓) → (𝜑 → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))))
10	8, 9	ax-mp 5	. . 3 ⊢ (((𝜑 → (𝜑 → 𝜒)) → (𝜑 → 𝜒)) → (((𝜑 → 𝜓) → (𝜑 → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
11	4, 7, 10	mpsyl 68	. 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
12	1, 11	tbsyl 34502	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 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: (None)