Theorem cbvopab2v 4636

Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)

Hypothesis

Ref	Expression
cbvopab2v.1	|- (y = z -> (ph <-> ps))

Assertion

Ref	Expression
cbvopab2v	|- {<.x, y>. | ph} = {<.x, z>. | ps}

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

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

Proof of Theorem cbvopab2v

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4579	. . . . . . 7 |- (y = z -> <.x, y>. = <.x, z>.)
2	1	eqeq2d 2364	. . . . . 6 |- (y = z -> (w = <.x, y>. <-> w = <.x, z>.))
3		cbvopab2v.1	. . . . . 6 |- (y = z -> (ph <-> ps))
4	2, 3	anbi12d 691	. . . . 5 |- (y = z -> ((w = <.x, y>. /\ ph) <-> (w = <.x, z>. /\ ps)))
5	4	cbvexv 2003	. . . 4 |- (E.y(w = <.x, y>. /\ ph) <-> E.z(w = <.x, z>. /\ ps))
6	5	exbii 1582	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.xE.z(w = <.x, z>. /\ ps))
7	6	abbii 2465	. 2 |- {w | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.xE.z(w = <.x, z>. /\ ps)}
8		df-opab 4623	. 2 |- {<.x, y>. | ph} = {w | E.xE.y(w = <.x, y>. /\ ph)}
9		df-opab 4623	. 2 |- {<.x, z>. | ps} = {w | E.xE.z(w = <.x, z>. /\ ps)}
10	7, 8, 9	3eqtr4i 2383	1 |- {<.x, y>. | ph} = {<.x, z>. | ps}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642  {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: (None)