Theorem ax16 1251

Description: Theorem showing that ax-16 1252 is redundant if ax-17 1012 is included in the axiom system. The important part of the proof is provided by aev 1250.

See ax16ALT 1313 for an alternate proof that does not require ax-10 1007 or ax-12 1009.

This theorem should not be referenced in any proof. Instead, use ax-16 1252 below so that theorems needing ax-16 1252 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 1250	. 2 |- (A.x x = y -> A.z x = z)
2		ax-17 1012	. . . 4 |- (ph -> A.zph)
3		sbequ12 1223	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
4	3	biimpcd 162	. . . 4 |- (ph -> (x = z -> [z / x]ph))
5	2, 4	19.20d 1037	. . 3 |- (ph -> (A.z x = z -> A.z[z / x]ph))
6	2	hbsb3 1248	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
7		stdpc7 1222	. . . 4 |- (z = x -> ([z / x]ph -> ph))
8	6, 2, 7	cbv3 1206	. . 3 |- (A.z[z / x]ph -> A.xph)
9	5, 8	syl6com 53	. 2 |- (A.z x = z -> (ph -> A.xph))
10	1, 9	syl 10	1 |- (A.x x = y -> (ph -> A.xph))

Syntax hints:   -> wi 3  A.wal 995   = wceq 997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-sb 1214

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