Theorem equsb3 1717

Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)

Assertion

Ref	Expression
equsb3	|- ([x / y]y = z <-> x = z)

Distinct variable group:   y,z

Proof of Theorem equsb3

Step	Hyp	Ref	Expression
1		equsb3lem 1715	. . 3 |- ([w / y]y = z <-> w = z)
2	1	sbbii 1538	. 2 |- ([x / w][w / y]y = z <-> [x / w]w = z)
3		ax-17 1317	. . 3 |- (y = z -> A.w y = z)
4	3	sbco2 1629	. 2 |- ([x / w][w / y]y = z <-> [x / y]y = z)
5		equsb3lem 1715	. 2 |- ([x / w]w = z <-> x = z)
6	2, 4, 5	3bitr3i 198	1 |- ([x / y]y = z <-> x = z)

Syntax hints:   <-> wb 163   = wceq 1298  [wsbc 1534

This theorem is referenced by:  sb8eu 1783  sbeqal1 16355

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-9 1307  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  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536

Copyright terms: Public domain