Theorem mpt-P1.8.2AC.2SH 59

Description: Another Deductive Form of mpt-P1.8 57.

Hypotheses

Ref	Expression
mpt-P1.8.2AC.2SH.1	⊢ (𝛾₁ → (𝛾₂ → 𝜑))
mpt-P1.8.2AC.2SH.2	⊢ (𝛾₁ → (𝛾₂ → (𝜑 → 𝜓)))

Assertion

Ref	Expression
mpt-P1.8.2AC.2SH	⊢ (𝛾₁ → (𝛾₂ → 𝜓))

Proof of Theorem mpt-P1.8.2AC.2SH

Step	Hyp	Ref	Expression
1		mpt-P1.8.2AC.2SH.2	. 2 ⊢ (𝛾₁ → (𝛾₂ → (𝜑 → 𝜓)))
2		mpt-P1.8.2AC.2SH.1	. . . 4 ⊢ (𝛾₁ → (𝛾₂ → 𝜑))
3		mpt-P1.8 57	. . . . . . 7 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
4	3	axL1.SH 30	. . . . . 6 ⊢ (𝛾₂ → (𝜑 → ((𝜑 → 𝜓) → 𝜓)))
5	4	axL1.SH 30	. . . . 5 ⊢ (𝛾₁ → (𝛾₂ → (𝜑 → ((𝜑 → 𝜓) → 𝜓))))
6	5	rcp-FR2.SH 42	. . . 4 ⊢ ((𝛾₁ → (𝛾₂ → 𝜑)) → (𝛾₁ → (𝛾₂ → ((𝜑 → 𝜓) → 𝜓))))
7	2, 6	ax-MP 14	. . 3 ⊢ (𝛾₁ → (𝛾₂ → ((𝜑 → 𝜓) → 𝜓)))
8	7	rcp-FR2.SH 42	. 2 ⊢ ((𝛾₁ → (𝛾₂ → (𝜑 → 𝜓))) → (𝛾₁ → (𝛾₂ → 𝜓)))
9	1, 8	ax-MP 14	1 ⊢ (𝛾₁ → (𝛾₂ → 𝜓))

Syntax hints:   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-MP 14

This theorem is referenced by:  bicmb-P2.5c  119  import-P2.10a  140