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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 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )

Proof

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