Metamath Proof Explorer


Theorem ax4fromc4

Description: Rederivation of Axiom ax-4 from ax-c4 , ax-c5 , ax-gen and minimal implicational calculus { ax-mp , ax-1 , ax-2 }. See axc4 for the derivation of ax-c4 from ax-4 . (Contributed by NM, 23-May-2008) (Proof modification is discouraged.) Use ax-4 instead. (New usage is discouraged.)

Ref Expression
Assertion ax4fromc4 ( ∀ 𝑥 ( 𝜑𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )

Proof

Step Hyp Ref Expression
1 ax-c4 ( ∀ 𝑥 ( ∀ 𝑥 ( 𝜑𝜓 ) → ( ∀ 𝑥 𝜑𝜓 ) ) → ( ∀ 𝑥 ( 𝜑𝜓 ) → ∀ 𝑥 ( ∀ 𝑥 𝜑𝜓 ) ) )
2 ax-c5 ( ∀ 𝑥 𝜑𝜑 )
3 ax-c5 ( ∀ 𝑥 ( 𝜑𝜓 ) → ( 𝜑𝜓 ) )
4 2 3 syl5 ( ∀ 𝑥 ( 𝜑𝜓 ) → ( ∀ 𝑥 𝜑𝜓 ) )
5 1 4 mpg ( ∀ 𝑥 ( 𝜑𝜓 ) → ∀ 𝑥 ( ∀ 𝑥 𝜑𝜓 ) )
6 ax-c4 ( ∀ 𝑥 ( ∀ 𝑥 𝜑𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
7 5 6 syl ( ∀ 𝑥 ( 𝜑𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )