Theorem cbvopab1s 2680

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution.

Assertion

Ref	Expression
cbvopab1s	|- {<.x, y>. | ph} = {<.z, y>. | [z / x]ph}

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

Proof of Theorem cbvopab1s

Step	Hyp	Ref	Expression
1		ax-17 973	. . . 4 |- (E.y(w = <.x, y>. /\ ph) -> A.zE.y(w = <.x, y>. /\ ph))
2		ax-17 973	. . . . . 6 |- (w = <.z, y>. -> A.x w = <.z, y>.)
3		hbs1 1334	. . . . . 6 |- ([z / x]ph -> A.x[z / x]ph)
4	2, 3	hban 1011	. . . . 5 |- ((w = <.z, y>. /\ [z / x]ph) -> A.x(w = <.z, y>. /\ [z / x]ph))
5	4	hbex 1008	. . . 4 |- (E.y(w = <.z, y>. /\ [z / x]ph) -> A.xE.y(w = <.z, y>. /\ [z / x]ph))
6		opeq1 2491	. . . . . . 7 |- (x = z -> <.x, y>. = <.z, y>.)
7	6	eqeq2d 1489	. . . . . 6 |- (x = z -> (w = <.x, y>. <-> w = <.z, y>.))
8		sbequ12 1183	. . . . . 6 |- (x = z -> (ph <-> [z / x]ph))
9	7, 8	anbi12d 630	. . . . 5 |- (x = z -> ((w = <.x, y>. /\ ph) <-> (w = <.z, y>. /\ [z / x]ph)))
10	9	exbidv 1281	. . . 4 |- (x = z -> (E.y(w = <.x, y>. /\ ph) <-> E.y(w = <.z, y>. /\ [z / x]ph)))
11	1, 5, 10	cbvex 1168	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.zE.y(w = <.z, y>. /\ [z / x]ph))
12	11	abbii 1578	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.zE.y(w = <.z, y>. /\ [z / x]ph)}
13		df-opab 2672	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
14		df-opab 2672	. 2 |- {<.z, y>. | [z / x]ph} = {w | E.zE.y(w = <.z, y>. /\ [z / x]ph)}
15	12, 13, 14	3eqtr4 1508	1 |- {<.x, y>. | ph} = {<.z, y>. | [z / x]ph}

Syntax hints:   /\ wa 223   = wceq 958  E.wex 982  [wsbc 1172  {cab 1466  <.cop 2415  {copab 2671

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-12 970  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672

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