Theorem ax16 1209

Description: Theorem showing that ax-16 1210 is redundant if ax-17 971 is included in the axiom system. The important part of the proof is provided by aev 1208.

See ax16ALT 1271 for an alternate proof that does not require ax-10 966 or ax-12 968.

This theorem should not be referenced in any proof. Instead, use ax-16 1210 below so that theorems needing ax-16 1210 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 1208	. 2 |- (A.x x = y -> A.z x = z)
2		ax-17 971	. . . 4 |- (ph -> A.zph)
3		sbequ12 1181	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
4	3	biimpcd 155	. . . 4 |- (ph -> (x = z -> [z / x]ph))
5	2, 4	19.20d 996	. . 3 |- (ph -> (A.z x = z -> A.z[z / x]ph))
6	2	hbsb3 1206	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
7		stdpc7 1180	. . . 4 |- (z = x -> ([z / x]ph -> ph))
8	6, 2, 7	cbv3 1164	. . 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 954   = wceq 956

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

Copyright terms: Public domain