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