Theorem trnsp-P1.15d.2AC.SH 85

Description: Another Deductive Form of trnsp-P1.15d 83.

Hypothesis

Ref	Expression
trnsp-P1.15d.2AC.SH.1	⊢ (𝛾₁ → (𝛾₂ → (¬ 𝜑 → ¬ 𝜓)))

Assertion

Ref	Expression
trnsp-P1.15d.2AC.SH	⊢ (𝛾₁ → (𝛾₂ → (𝜓 → 𝜑)))

Proof of Theorem trnsp-P1.15d.2AC.SH

Step	Hyp	Ref	Expression
1		trnsp-P1.15d.2AC.SH.1	. 2 ⊢ (𝛾₁ → (𝛾₂ → (¬ 𝜑 → ¬ 𝜓)))
2		trnsp-P1.15d 83	. . . . 5 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
3	2	axL1.SH 30	. . . 4 ⊢ (𝛾₂ → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
4	3	axL1.SH 30	. . 3 ⊢ (𝛾₁ → (𝛾₂ → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))))
5	4	rcp-FR2.SH 42	. 2 ⊢ ((𝛾₁ → (𝛾₂ → (¬ 𝜑 → ¬ 𝜓))) → (𝛾₁ → (𝛾₂ → (𝜓 → 𝜑))))
6	1, 5	ax-MP 14	1 ⊢ (𝛾₁ → (𝛾₂ → (𝜓 → 𝜑)))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem is referenced by:  export-L2.1b  93