Theorem retbwax2

Description: tbw-ax2 rederived from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	retbwax2	⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		merco1lem1	⊢ ( ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ⊥ → 𝜑 ) )
2		merco1	⊢ ( ( ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ⊥ → 𝜑 ) ) → ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) )
4		merco1	⊢ ( ( ( ( ( 𝜑 → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ⊥ ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) )
5		merco1	⊢ ( ( ( ( ( ( 𝜑 → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ⊥ ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) ) → ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) )
6	4 5	ax-mp	⊢ ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
7	3 6	ax-mp	⊢ ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) )
8		merco1lem1	⊢ ( ( ( ( ( 𝜓 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ⊥ → 𝜑 ) )
9		merco1	⊢ ( ( ( ( ( ( 𝜓 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ⊥ → 𝜑 ) ) → ( ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) )
10	8 9	ax-mp	⊢ ( ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) )
11		merco1	⊢ ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → ⊥ ) ) → ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ⊥ ) ) → ⊥ ) → ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) )
12		merco1	⊢ ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → ⊥ ) ) → ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ⊥ ) ) → ⊥ ) → ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) ) → ( ( ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) ) )
13	11 12	ax-mp	⊢ ( ( ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) )
14	10 13	ax-mp	⊢ ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) )
15	7 14	ax-mp	⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )