Metamath Proof Explorer


Theorem re1ax2

Description: ax-2 rederived from the Tarski-Bernays axiom system. Often tb-ax1 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.)

Ref Expression
Assertion re1ax2 ( ( 𝜑 → ( 𝜓𝜒 ) ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 re1ax2lem ( ( 𝜑 → ( 𝜓𝜒 ) ) → ( 𝜓 → ( 𝜑𝜒 ) ) )
2 tb-ax1 ( ( 𝜑 → ( 𝜑𝜒 ) ) → ( ( ( 𝜑𝜒 ) → 𝜒 ) → ( 𝜑𝜒 ) ) )
3 tb-ax3 ( ( ( ( 𝜑𝜒 ) → 𝜒 ) → ( 𝜑𝜒 ) ) → ( 𝜑𝜒 ) )
4 2 3 tbsyl ( ( 𝜑 → ( 𝜑𝜒 ) ) → ( 𝜑𝜒 ) )
5 tb-ax1 ( ( 𝜑𝜓 ) → ( ( 𝜓 → ( 𝜑𝜒 ) ) → ( 𝜑 → ( 𝜑𝜒 ) ) ) )
6 re1ax2lem ( ( ( 𝜑𝜓 ) → ( ( 𝜓 → ( 𝜑𝜒 ) ) → ( 𝜑 → ( 𝜑𝜒 ) ) ) ) → ( ( 𝜓 → ( 𝜑𝜒 ) ) → ( ( 𝜑𝜓 ) → ( 𝜑 → ( 𝜑𝜒 ) ) ) ) )
7 5 6 ax-mp ( ( 𝜓 → ( 𝜑𝜒 ) ) → ( ( 𝜑𝜓 ) → ( 𝜑 → ( 𝜑𝜒 ) ) ) )
8 tb-ax1 ( ( ( 𝜑𝜓 ) → ( 𝜑 → ( 𝜑𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜑𝜒 ) ) → ( 𝜑𝜒 ) ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) ) )
9 re1ax2lem ( ( ( ( 𝜑𝜓 ) → ( 𝜑 → ( 𝜑𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜑𝜒 ) ) → ( 𝜑𝜒 ) ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) ) ) → ( ( ( 𝜑 → ( 𝜑𝜒 ) ) → ( 𝜑𝜒 ) ) → ( ( ( 𝜑𝜓 ) → ( 𝜑 → ( 𝜑𝜒 ) ) ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) ) ) )
10 8 9 ax-mp ( ( ( 𝜑 → ( 𝜑𝜒 ) ) → ( 𝜑𝜒 ) ) → ( ( ( 𝜑𝜓 ) → ( 𝜑 → ( 𝜑𝜒 ) ) ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) ) )
11 4 7 10 mpsyl ( ( 𝜓 → ( 𝜑𝜒 ) ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) )
12 1 11 tbsyl ( ( 𝜑 → ( 𝜓𝜒 ) ) → ( ( 𝜑𝜓 ) → ( 𝜑𝜒 ) ) )