Users' Mathboxes Mathbox for Anthony Hart < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  re1ax2 Structured version   Visualization version   GIF version

Theorem re1ax2 32508
Description: ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 32503 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 32507 . 2 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
2 tb-ax1 32503 . . . 4 ((𝜑 → (𝜑𝜒)) → (((𝜑𝜒) → 𝜒) → (𝜑𝜒)))
3 tb-ax3 32505 . . . 4 ((((𝜑𝜒) → 𝜒) → (𝜑𝜒)) → (𝜑𝜒))
42, 3tbsyl 32506 . . 3 ((𝜑 → (𝜑𝜒)) → (𝜑𝜒))
5 tb-ax1 32503 . . . 4 ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜑 → (𝜑𝜒))))
6 re1ax2lem 32507 . . . 4 (((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜑 → (𝜑𝜒)))) → ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑 → (𝜑𝜒)))))
75, 6ax-mp 5 . . 3 ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑 → (𝜑𝜒))))
8 tb-ax1 32503 . . . 4 (((𝜑𝜓) → (𝜑 → (𝜑𝜒))) → (((𝜑 → (𝜑𝜒)) → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒))))
9 re1ax2lem 32507 . . . 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 68 . 2 ((𝜓 → (𝜑𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
121, 11tbsyl 32506 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)
  Copyright terms: Public domain W3C validator