Theorem ax16 1579

Description: Theorem showing that ax-16 1580 is redundant if ax-17 1317 is included in the axiom system. The important part of the proof is provided by aev 1577.

See ax16ALT 1648 for an alternate proof that does not require ax-10 1308 or ax-12 1310.

This theorem should not be referenced in any proof. Instead, use ax-16 1580 below so that theorems needing ax-16 1580 can be more easily identified.

Assertion

Ref	Expression
ax16	|- (A.x x = y -> (ph -> A.xph))

Distinct variable group:   x,y

Proof of Theorem ax16

Step	Hyp	Ref	Expression
1		aev 1577	. 2 |- (A.x x = y -> A.z x = z)
2		ax-17 1317	. . . 4 |- (ph -> A.zph)
3		sbequ12 1545	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
4	3	biimpcd 172	. . . 4 |- (ph -> (x = z -> [z / x]ph))
5	2, 4	alimd 1343	. . 3 |- (ph -> (A.z x = z -> A.z[z / x]ph))
6	2	hbsb3 1575	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
7		stdpc7 1544	. . . 4 |- (z = x -> ([z / x]ph -> ph))
8	6, 2, 7	cbv3 1525	. . 3 |- (A.z[z / x]ph -> A.xph)
9	5, 8	syl6com 64	. 2 |- (A.z x = z -> (ph -> A.xph))
10	1, 9	syl 12	1 |- (A.x x = y -> (ph -> A.xph))

Syntax hints:   -> wi 3  A.wal 1296   = wceq 1298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536

Copyright terms: Public domain