Theorem ax16 1211

Description: Theorem showing that ax-16 1212 is redundant if ax-17 973 is included in the axiom system. The important part of the proof is provided by aev 1210.

See ax16ALT 1273 for an alternate proof that does not require ax-10 968 or ax-12 970.

This theorem should not be referenced in any proof. Instead, use ax-16 1212 below so that theorems needing ax-16 1212 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 1210	. 2 |- (A.x x = y -> A.z x = z)
2		ax-17 973	. . . 4 |- (ph -> A.zph)
3		sbequ12 1183	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
4	3	biimpcd 155	. . . 4 |- (ph -> (x = z -> [z / x]ph))
5	2, 4	19.20d 998	. . 3 |- (ph -> (A.z x = z -> A.z[z / x]ph))
6	2	hbsb3 1208	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
7		stdpc7 1182	. . . 4 |- (z = x -> ([z / x]ph -> ph))
8	6, 2, 7	cbv3 1166	. . 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 956   = wceq 958

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174

Copyright terms: Public domain