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 φ ψ x φ x ψ

Proof

Step Hyp Ref Expression
1 ax-c4 x x φ ψ x φ ψ x φ ψ x x φ ψ
2 ax-c5 x φ φ
3 ax-c5 x φ ψ φ ψ
4 2 3 syl5 x φ ψ x φ ψ
5 1 4 mpg x φ ψ x x φ ψ
6 ax-c4 x x φ ψ x φ x ψ
7 5 6 syl x φ ψ x φ x ψ