Theorem cbvopab2 4634

Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Hypotheses

Ref	Expression
cbvopab2.1	|- F/zph
cbvopab2.2	|- F/yps
cbvopab2.3	|- (y = z -> (ph <-> ps))

Assertion

Ref	Expression
cbvopab2	|- {<.x, y>. | ph} = {<.x, z>. | ps}

Distinct variable group:   x,y,z

Allowed substitution hints:   ph(x,y,z)   ps(x,y,z)

Proof of Theorem cbvopab2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . . 6 |- F/z w = <.x, y>.
2		cbvopab2.1	. . . . . 6 |- F/zph
3	1, 2	nfan 1824	. . . . 5 |- F/z(w = <.x, y>. /\ ph)
4		nfv 1619	. . . . . 6 |- F/y w = <.x, z>.
5		cbvopab2.2	. . . . . 6 |- F/yps
6	4, 5	nfan 1824	. . . . 5 |- F/y(w = <.x, z>. /\ ps)
7		opeq2 4580	. . . . . . 7 |- (y = z -> <.x, y>. = <.x, z>.)
8	7	eqeq2d 2364	. . . . . 6 |- (y = z -> (w = <.x, y>. <-> w = <.x, z>.))
9		cbvopab2.3	. . . . . 6 |- (y = z -> (ph <-> ps))
10	8, 9	anbi12d 691	. . . . 5 |- (y = z -> ((w = <.x, y>. /\ ph) <-> (w = <.x, z>. /\ ps)))
11	3, 6, 10	cbvex 1985	. . . 4 |- (E.y(w = <.x, y>. /\ ph) <-> E.z(w = <.x, z>. /\ ps))
12	11	exbii 1582	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.xE.z(w = <.x, z>. /\ ps))
13	12	abbii 2466	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.xE.z(w = <.x, z>. /\ ps)}
14		df-opab 4624	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
15		df-opab 4624	. 2 |- {<.x, z>. | ps} = {w | E.xE.z(w = <.x, z>. /\ ps)}
16	13, 14, 15	3eqtr4i 2383	1 |- {<.x, y>. | ph} = {<.x, z>. | ps}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642  {cab 2339  <.cop 4562  {copab 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624

This theorem is referenced by:  cbvoprab3  5572