Theorem rcp-FR3 43

Description: Frege Axiom with 3 Antecedents.

Assertion

Ref	Expression
rcp-FR3	⊢ ((𝛾₃ → (𝛾₂ → (𝛾₁ → (𝜑 → 𝜓)))) → ((𝛾₃ → (𝛾₂ → (𝛾₁ → 𝜑))) → (𝛾₃ → (𝛾₂ → (𝛾₁ → 𝜓)))))

Proof of Theorem rcp-FR3

Step	Hyp	Ref	Expression
1		rcp-FR2 41	. . . 4 ⊢ ((𝛾₂ → (𝛾₁ → (𝜑 → 𝜓))) → ((𝛾₂ → (𝛾₁ → 𝜑)) → (𝛾₂ → (𝛾₁ → 𝜓))))
2	1	axL1.SH 30	. . 3 ⊢ (𝛾₃ → ((𝛾₂ → (𝛾₁ → (𝜑 → 𝜓))) → ((𝛾₂ → (𝛾₁ → 𝜑)) → (𝛾₂ → (𝛾₁ → 𝜓)))))
3	2	axL2.SH 31	. 2 ⊢ ((𝛾₃ → (𝛾₂ → (𝛾₁ → (𝜑 → 𝜓)))) → (𝛾₃ → ((𝛾₂ → (𝛾₁ → 𝜑)) → (𝛾₂ → (𝛾₁ → 𝜓)))))
4		ax-L2 12	. 2 ⊢ ((𝛾₃ → ((𝛾₂ → (𝛾₁ → 𝜑)) → (𝛾₂ → (𝛾₁ → 𝜓)))) → ((𝛾₃ → (𝛾₂ → (𝛾₁ → 𝜑))) → (𝛾₃ → (𝛾₂ → (𝛾₁ → 𝜓)))))
5	3, 4	syl-P1.2 34	1 ⊢ ((𝛾₃ → (𝛾₂ → (𝛾₁ → (𝜑 → 𝜓)))) → ((𝛾₃ → (𝛾₂ → (𝛾₁ → 𝜑))) → (𝛾₃ → (𝛾₂ → (𝛾₁ → 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:  rcp-FR3.SH  44