Theorem rcp-FR2 41

Description: Frege Axiom with Two Antecedents.

Assertion

Ref	Expression
rcp-FR2	⊢ ((𝛾₂ → (𝛾₁ → (𝜑 → 𝜓))) → ((𝛾₂ → (𝛾₁ → 𝜑)) → (𝛾₂ → (𝛾₁ → 𝜓))))

Proof of Theorem rcp-FR2

Step	Hyp	Ref	Expression
1		rcp-FR1 39	. . . 4 ⊢ ((𝛾₁ → (𝜑 → 𝜓)) → ((𝛾₁ → 𝜑) → (𝛾₁ → 𝜓)))
2	1	axL1.SH 30	. . 3 ⊢ (𝛾₂ → ((𝛾₁ → (𝜑 → 𝜓)) → ((𝛾₁ → 𝜑) → (𝛾₁ → 𝜓))))
3	2	axL2.SH 31	. 2 ⊢ ((𝛾₂ → (𝛾₁ → (𝜑 → 𝜓))) → (𝛾₂ → ((𝛾₁ → 𝜑) → (𝛾₁ → 𝜓))))
4		ax-L2 12	. 2 ⊢ ((𝛾₂ → ((𝛾₁ → 𝜑) → (𝛾₁ → 𝜓))) → ((𝛾₂ → (𝛾₁ → 𝜑)) → (𝛾₂ → (𝛾₁ → 𝜓))))
5	3, 4	syl-P1.2 34	1 ⊢ ((𝛾₂ → (𝛾₁ → (𝜑 → 𝜓))) → ((𝛾₂ → (𝛾₁ → 𝜑)) → (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-FR2.SH  42  rcp-FR3  43