Theorem isarep1 5360

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 2775	. . 3 |- b e. _V
2	1	elima 5027	. 2 |- ( b e. ( { <. x , y >. | ph } " A ) <-> E. z e. A z { <. x , y >. | ph } b )
3		df-br 4045	. . . 4 |- ( z { <. x , y >. | ph } b <-> <. z , b >. e. { <. x , y >. | ph } )
4		opelopabsb 4306	. . . 4 |- ( <. z , b >. e. { <. x , y >. | ph } <-> [. z / x ]. [. b / y ]. ph )
5		sbsbc 3002	. . . . . 6 |- ( [ b / y ] ph <-> [. b / y ]. ph )
6	5	sbbii 1788	. . . . 5 |- ( [ z / x ] [ b / y ] ph <-> [ z / x ] [. b / y ]. ph )
7		sbsbc 3002	. . . . 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 2513	. 2 |- ( E. z e. A z { <. x , y >. | ph } b <-> E. z e. A [ z / x ] [ b / y ] ph )
11		nfs1v 1967	. . 3 |- F/ x [ z / x ] [ b / y ] ph
12		nfv 1551	. . 3 |- F/ z [ b / y ] ph
13		sbequ12r 1795	. . 3 |- ( z = x -> ( [ z / x ] [ b / y ] ph <-> [ b / y ] ph ) )
14	11, 12, 13	cbvrex 2735	. 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 1785   e. wcel 2176   E.wrex 2485   [.wsbc 2998   <.cop 3636   class class class wbr 4044   {copab 4104   "cima 4678

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by: (None)