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Theorem re1ax2 31553
 Description: ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 31548 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 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Proof of Theorem re1ax2
StepHypRef Expression
1 re1ax2lem 31552 . 2 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
2 tb-ax1 31548 . . . 4 ((𝜑 → (𝜑𝜒)) → (((𝜑𝜒) → 𝜒) → (𝜑𝜒)))
3 tb-ax3 31550 . . . 4 ((((𝜑𝜒) → 𝜒) → (𝜑𝜒)) → (𝜑𝜒))
42, 3tbsyl 31551 . . 3 ((𝜑 → (𝜑𝜒)) → (𝜑𝜒))
5 tb-ax1 31548 . . . 4 ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜑 → (𝜑𝜒))))
6 re1ax2lem 31552 . . . 4 (((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜑 → (𝜑𝜒)))) → ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑 → (𝜑𝜒)))))
75, 6ax-mp 5 . . 3 ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑 → (𝜑𝜒))))
8 tb-ax1 31548 . . . 4 (((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → (((𝜑 → (𝜑𝜒)) → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒))))
9 re1ax2lem 31552 . . . 4 ((((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → (((𝜑 → (𝜑𝜒)) → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))) → (((𝜑 → (𝜑𝜒)) → (𝜑𝜒)) → (((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒)))))
108, 9ax-mp 5 . . 3 (((𝜑 → (𝜑𝜒)) → (𝜑𝜒)) → (((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → ((𝜑𝜓) → (𝜑𝜒))))
114, 7, 10mpsyl 66 . 2 ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
121, 11tbsyl 31551 1 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 Colors of variables: wff setvar class 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)
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