Theorem cbvoprab1 5567

Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)

Hypotheses

Ref	Expression
cbvoprab1.1	|- F/wph
cbvoprab1.2	|- F/xps
cbvoprab1.3	|- (x = w -> (ph <-> ps))

Assertion

Ref	Expression
cbvoprab1	|- {<.<.x, y>., z>. | ph} = {<.<.w, y>., z>. | ps}

Distinct variable group:   x,y,z,w

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

Proof of Theorem cbvoprab1

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . . 6 |- F/w v = <.x, y>.
2		cbvoprab1.1	. . . . . 6 |- F/wph
3	1, 2	nfan 1824	. . . . 5 |- F/w(v = <.x, y>. /\ ph)
4	3	nfex 1843	. . . 4 |- F/wE.y(v = <.x, y>. /\ ph)
5		nfv 1619	. . . . . 6 |- F/x v = <.w, y>.
6		cbvoprab1.2	. . . . . 6 |- F/xps
7	5, 6	nfan 1824	. . . . 5 |- F/x(v = <.w, y>. /\ ps)
8	7	nfex 1843	. . . 4 |- F/xE.y(v = <.w, y>. /\ ps)
9		opeq1 4578	. . . . . . 7 |- (x = w -> <.x, y>. = <.w, y>.)
10	9	eqeq2d 2364	. . . . . 6 |- (x = w -> (v = <.x, y>. <-> v = <.w, y>.))
11		cbvoprab1.3	. . . . . 6 |- (x = w -> (ph <-> ps))
12	10, 11	anbi12d 691	. . . . 5 |- (x = w -> ((v = <.x, y>. /\ ph) <-> (v = <.w, y>. /\ ps)))
13	12	exbidv 1626	. . . 4 |- (x = w -> (E.y(v = <.x, y>. /\ ph) <-> E.y(v = <.w, y>. /\ ps)))
14	4, 8, 13	cbvex 1985	. . 3 |- (E.xE.y(v = <.x, y>. /\ ph) <-> E.wE.y(v = <.w, y>. /\ ps))
15	14	opabbii 4626	. 2 |- {<.v, z>. | E.xE.y(v = <.x, y>. /\ ph)} = {<.v, z>. | E.wE.y(v = <.w, y>. /\ ps)}
16		dfoprab2 5558	. 2 |- {<.<.x, y>., z>. | ph} = {<.v, z>. | E.xE.y(v = <.x, y>. /\ ph)}
17		dfoprab2 5558	. 2 |- {<.<.w, y>., z>. | ps} = {<.v, z>. | E.wE.y(v = <.w, y>. /\ ps)}
18	15, 16, 17	3eqtr4i 2383	1 |- {<.<.x, y>., z>. | ph} = {<.<.w, y>., z>. | ps}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642  <.cop 4561  {copab 4622  {coprab 5527

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-oprab 5528

This theorem is referenced by: (None)