Theorem isarep1 5299

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 2740	. . 3 |- b e. _V
2	1	elima 4972	. 2 |- ( b e. ( { <. x , y >. | ph } " A ) <-> E. z e. A z { <. x , y >. | ph } b )
3		df-br 4002	. . . 4 |- ( z { <. x , y >. | ph } b <-> <. z , b >. e. { <. x , y >. | ph } )
4		opelopabsb 4258	. . . 4 |- ( <. z , b >. e. { <. x , y >. | ph } <-> [. z / x ]. [. b / y ]. ph )
5		sbsbc 2966	. . . . . 6 |- ( [ b / y ] ph <-> [. b / y ]. ph )
6	5	sbbii 1765	. . . . 5 |- ( [ z / x ] [ b / y ] ph <-> [ z / x ] [. b / y ]. ph )
7		sbsbc 2966	. . . . 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 2484	. 2 |- ( E. z e. A z { <. x , y >. | ph } b <-> E. z e. A [ z / x ] [ b / y ] ph )
11		nfs1v 1939	. . 3 |- F/ x [ z / x ] [ b / y ] ph
12		nfv 1528	. . 3 |- F/ z [ b / y ] ph
13		sbequ12r 1772	. . 3 |- ( z = x -> ( [ z / x ] [ b / y ] ph <-> [ b / y ] ph ) )
14	11, 12, 13	cbvrex 2700	. 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 1762   e. wcel 2148   E.wrex 2456   [.wsbc 2962   <.cop 3595   class class class wbr 4001   {copab 4061   "cima 4627

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4002  df-opab 4063  df-xp 4630  df-cnv 4632  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637

This theorem is referenced by: (None)