Theorem brabsb 2822

Description: The law of concretion in terms of substitutions.

Hypothesis

Ref	Expression
brabsb.1	|- R = {<.x, y>. | ph}

Assertion

Ref	Expression
brabsb	|- (zRw <-> [w / y][z / x]ph)

Distinct variable groups:   x,y,z   x,w,y

Proof of Theorem brabsb

Step	Hyp	Ref	Expression
1		brabsb.1	. . 3 |- R = {<.x, y>. | ph}
2	1	breqi 2630	. 2 |- (zRw <-> z{<.x, y>. | ph}w)
3		df-br 2625	. 2 |- (z{<.x, y>. | ph}w <-> <.z, w>. e. {<.x, y>. | ph})
4		opabsb 2821	. 2 |- (<.z, w>. e. {<.x, y>. | ph} <-> [w / y][z / x]ph)
5	2, 3, 4	3bitr 177	1 |- (zRw <-> [w / y][z / x]ph)

Syntax hints:   <-> wb 146   = wceq 958   e. wcel 960  [wsbc 1172  <.cop 2415   class class class wbr 2624  {copab 2671

This theorem is referenced by:  eqerlem 4276

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-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672

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