Theorem cbvopab1 4632

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
cbvopab1.1	|- F/zph
cbvopab1.2	|- F/xps
cbvopab1.3	|- (x = z -> (ph <-> ps))

Assertion

Ref	Expression
cbvopab1	|- {<.x, y>. | ph} = {<.z, y>. | ps}

Distinct variable groups:   x,y   y,z

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

Proof of Theorem cbvopab1

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . 5 |- F/vE.y(w = <.x, y>. /\ ph)
2		nfv 1619	. . . . . . 7 |- F/x w = <.v, y>.
3		nfs1v 2106	. . . . . . 7 |- F/x[v / x]ph
4	2, 3	nfan 1824	. . . . . 6 |- F/x(w = <.v, y>. /\ [v / x]ph)
5	4	nfex 1843	. . . . 5 |- F/xE.y(w = <.v, y>. /\ [v / x]ph)
6		opeq1 4578	. . . . . . . 8 |- (x = v -> <.x, y>. = <.v, y>.)
7	6	eqeq2d 2364	. . . . . . 7 |- (x = v -> (w = <.x, y>. <-> w = <.v, y>.))
8		sbequ12 1919	. . . . . . 7 |- (x = v -> (ph <-> [v / x]ph))
9	7, 8	anbi12d 691	. . . . . 6 |- (x = v -> ((w = <.x, y>. /\ ph) <-> (w = <.v, y>. /\ [v / x]ph)))
10	9	exbidv 1626	. . . . 5 |- (x = v -> (E.y(w = <.x, y>. /\ ph) <-> E.y(w = <.v, y>. /\ [v / x]ph)))
11	1, 5, 10	cbvex 1985	. . . 4 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.vE.y(w = <.v, y>. /\ [v / x]ph))
12		nfv 1619	. . . . . . 7 |- F/z w = <.v, y>.
13		cbvopab1.1	. . . . . . . 8 |- F/zph
14	13	nfsb 2109	. . . . . . 7 |- F/z[v / x]ph
15	12, 14	nfan 1824	. . . . . 6 |- F/z(w = <.v, y>. /\ [v / x]ph)
16	15	nfex 1843	. . . . 5 |- F/zE.y(w = <.v, y>. /\ [v / x]ph)
17		nfv 1619	. . . . 5 |- F/vE.y(w = <.z, y>. /\ ps)
18		opeq1 4578	. . . . . . . 8 |- (v = z -> <.v, y>. = <.z, y>.)
19	18	eqeq2d 2364	. . . . . . 7 |- (v = z -> (w = <.v, y>. <-> w = <.z, y>.))
20		sbequ 2060	. . . . . . . 8 |- (v = z -> ([v / x]ph <-> [z / x]ph))
21		cbvopab1.2	. . . . . . . . 9 |- F/xps
22		cbvopab1.3	. . . . . . . . 9 |- (x = z -> (ph <-> ps))
23	21, 22	sbie 2038	. . . . . . . 8 |- ([z / x]ph <-> ps)
24	20, 23	syl6bb 252	. . . . . . 7 |- (v = z -> ([v / x]ph <-> ps))
25	19, 24	anbi12d 691	. . . . . 6 |- (v = z -> ((w = <.v, y>. /\ [v / x]ph) <-> (w = <.z, y>. /\ ps)))
26	25	exbidv 1626	. . . . 5 |- (v = z -> (E.y(w = <.v, y>. /\ [v / x]ph) <-> E.y(w = <.z, y>. /\ ps)))
27	16, 17, 26	cbvex 1985	. . . 4 |- (E.vE.y(w = <.v, y>. /\ [v / x]ph) <-> E.zE.y(w = <.z, y>. /\ ps))
28	11, 27	bitri 240	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.zE.y(w = <.z, y>. /\ ps))
29	28	abbii 2465	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.zE.y(w = <.z, y>. /\ ps)}
30		df-opab 4623	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
31		df-opab 4623	. 2 |- {<.z, y>. | ps} = {w | E.zE.y(w = <.z, y>. /\ ps)}
32	29, 30, 31	3eqtr4i 2383	1 |- {<.x, y>. | ph} = {<.z, y>. | ps}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642  [wsb 1648  {cab 2339  <.cop 4561  {copab 4622

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

This theorem is referenced by:  cbvopab1v  4635  cbvmpt  5676