Theorem brabsb 2816

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 2625	. 2 |- (zRw <-> z{<.x, y>. | ph}w)
3		df-br 2620	. 2 |- (z{<.x, y>. | ph}w <-> <.z, w>. e. {<.x, y>. | ph})
4		opabsb 2815	. 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 956   e. wcel 958  [wsbc 1170  <.cop 2411   class class class wbr 2619  {copab 2666

This theorem is referenced by:  eqerlem 4270

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-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667

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