Theorem isarep1 5416

Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by ph ( x , y ) i.e. the class ( { <. x , y >. | ph } " A ). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
isarep1	|- ( b e. ( { <. x , y >. | ph } " A ) <-> E. x e. A [ b / y ] ph )

Distinct variable groups:   x, A   x, b, y

Allowed substitution hints:   ph( x, y, b)   A( y, b)

Proof of Theorem isarep1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2805	. . 3 |- b e. _V
2	1	elima 5081	. 2 |- ( b e. ( { <. x , y >. | ph } " A ) <-> E. z e. A z { <. x , y >. | ph } b )
3		df-br 4089	. . . 4 |- ( z { <. x , y >. | ph } b <-> <. z , b >. e. { <. x , y >. | ph } )
4		opelopabsb 4354	. . . 4 |- ( <. z , b >. e. { <. x , y >. | ph } <-> [. z / x ]. [. b / y ]. ph )
5		sbsbc 3035	. . . . . 6 |- ( [ b / y ] ph <-> [. b / y ]. ph )
6	5	sbbii 1813	. . . . 5 |- ( [ z / x ] [ b / y ] ph <-> [ z / x ] [. b / y ]. ph )
7		sbsbc 3035	. . . . 5 |- ( [ z / x ] [. b / y ]. ph <-> [. z / x ]. [. b / y ]. ph )
8	6, 7	bitr2i 185	. . . 4 |- ( [. z / x ]. [. b / y ]. ph <-> [ z / x ] [ b / y ] ph )
9	3, 4, 8	3bitri 206	. . 3 |- ( z { <. x , y >. | ph } b <-> [ z / x ] [ b / y ] ph )
10	9	rexbii 2539	. 2 |- ( E. z e. A z { <. x , y >. | ph } b <-> E. z e. A [ z / x ] [ b / y ] ph )
11		nfs1v 1992	. . 3 |- F/ x [ z / x ] [ b / y ] ph
12		nfv 1576	. . 3 |- F/ z [ b / y ] ph
13		sbequ12r 1820	. . 3 |- ( z = x -> ( [ z / x ] [ b / y ] ph <-> [ b / y ] ph ) )
14	11, 12, 13	cbvrex 2764	. 2 |- ( E. z e. A [ z / x ] [ b / y ] ph <-> E. x e. A [ b / y ] ph )
15	2, 10, 14	3bitri 206	1 |- ( b e. ( { <. x , y >. | ph } " A ) <-> E. x e. A [ b / y ] ph )

Syntax hints:   <-> wb 105   [wsb 1810   e. wcel 2202   E.wrex 2511   [.wsbc 3031   <.cop 3672   class class class wbr 4088   {copab 4149   "cima 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by: (None)