Theorem mpt-P1.8.3AC.2SH 60

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

Hypotheses

Ref	Expression
mpt-P1.8.3AC.2SH.1	⊢ (𝛾₁ → (𝛾₂ → (𝛾₃ → 𝜑)))
mpt-P1.8.3AC.2SH.2	⊢ (𝛾₁ → (𝛾₂ → (𝛾₃ → (𝜑 → 𝜓))))

Assertion

Ref	Expression
mpt-P1.8.3AC.2SH	⊢ (𝛾₁ → (𝛾₂ → (𝛾₃ → 𝜓)))

Proof of Theorem mpt-P1.8.3AC.2SH

Step	Hyp	Ref	Expression
1		mpt-P1.8.3AC.2SH.2	. 2 ⊢ (𝛾₁ → (𝛾₂ → (𝛾₃ → (𝜑 → 𝜓))))
2		mpt-P1.8.3AC.2SH.1	. . . 4 ⊢ (𝛾₁ → (𝛾₂ → (𝛾₃ → 𝜑)))
3		mpt-P1.8 57	. . . . . . . 8 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
4	3	axL1.SH 30	. . . . . . 7 ⊢ (𝛾₃ → (𝜑 → ((𝜑 → 𝜓) → 𝜓)))
5	4	axL1.SH 30	. . . . . 6 ⊢ (𝛾₂ → (𝛾₃ → (𝜑 → ((𝜑 → 𝜓) → 𝜓))))
6	5	axL1.SH 30	. . . . 5 ⊢ (𝛾₁ → (𝛾₂ → (𝛾₃ → (𝜑 → ((𝜑 → 𝜓) → 𝜓)))))
7	6	rcp-FR3.SH 44	. . . 4 ⊢ ((𝛾₁ → (𝛾₂ → (𝛾₃ → 𝜑))) → (𝛾₁ → (𝛾₂ → (𝛾₃ → ((𝜑 → 𝜓) → 𝜓)))))
8	2, 7	ax-MP 14	. . 3 ⊢ (𝛾₁ → (𝛾₂ → (𝛾₃ → ((𝜑 → 𝜓) → 𝜓))))
9	8	rcp-FR3.SH 44	. 2 ⊢ ((𝛾₁ → (𝛾₂ → (𝛾₃ → (𝜑 → 𝜓)))) → (𝛾₁ → (𝛾₂ → (𝛾₃ → 𝜓))))
10	1, 9	ax-MP 14	1 ⊢ (𝛾₁ → (𝛾₂ → (𝛾₃ → 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:  export-P2.10b  142