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 ) ) )