Theorem ax5 1017

Description: Rederivation of axiom ax-5 1001 from the orginal version, ax-5o 1016. See ax5o 1015 for the derivation of ax-5o 1016 from ax-5 1001.

This theorem should not be referenced in any proof. Instead, use ax-5 1001 above so that uses of ax-5 1001 can be more easily identified.

Assertion

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

Proof of Theorem ax5

Step	Hyp	Ref	Expression
1		ax-4 1014	. . . . 5 |- (A.x(ph -> ps) -> (ph -> ps))
2		ax-4 1014	. . . . 5 |- (A.xph -> ph)
3	1, 2	syl5 21	. . . 4 |- (A.x(ph -> ps) -> (A.xph -> ps))
4	3	ax-gen 1004	. . 3 |- A.x(A.x(ph -> ps) -> (A.xph -> ps))
5		ax-5o 1016	. . 3 |- (A.x(A.x(ph -> ps) -> (A.xph -> ps)) -> (A.x(ph -> ps) -> A.x(A.xph -> ps)))
6	4, 5	ax-mp 7	. 2 |- (A.x(ph -> ps) -> A.x(A.xph -> ps))
7		ax-5o 1016	. 2 |- (A.x(A.xph -> ps) -> (A.xph -> A.xps))
8	6, 7	syl 10	1 |- (A.x(ph -> ps) -> (A.xph -> A.xps))

Syntax hints:   -> wi 3  A.wal 995

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-mp 7  ax-gen 1004  ax-4 1014  ax-5o 1016

Copyright terms: Public domain