Theorem ax5 2146

Description: Rederivation of axiom ax-5 1557 from ax-5o 2136 and other older axioms. See ax5o 1749 for the derivation of ax-5o 2136 from ax-5 1557. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax5	|- (A.x(ph -> ps) -> (A.xph -> A.xps))

Proof of Theorem ax5

Step	Hyp	Ref	Expression
1		ax-5o 2136	. . 3 |- (A.x(A.x(ph -> ps) -> (A.xph -> ps)) -> (A.x(ph -> ps) -> A.x(A.xph -> ps)))
2		ax-4 2135	. . . 4 |- (A.xph -> ph)
3		ax-4 2135	. . . 4 |- (A.x(ph -> ps) -> (ph -> ps))
4	2, 3	syl5 28	. . 3 |- (A.x(ph -> ps) -> (A.xph -> ps))
5	1, 4	mpg 1548	. 2 |- (A.x(ph -> ps) -> A.x(A.xph -> ps))
6		ax-5o 2136	. 2 |- (A.x(A.xph -> ps) -> (A.xph -> A.xps))
7	5, 6	syl 15	1 |- (A.x(ph -> ps) -> (A.xph -> A.xps))

Syntax hints:   -> wi 4  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1546  ax-4 2135  ax-5o 2136

This theorem is referenced by: (None)