Theorem abrexex2OLD 7150

Description: Obsolete proof of abrexex2 7148 as of 8-Dec-2021. (Contributed by NM, 12-Sep-2004.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
abrexex2OLD.1	|- A e. _V
abrexex2OLD.2	|- { y | ph } e. _V

Assertion

Ref	Expression
abrexex2OLD	|- { y | E. x e. A ph } e. _V

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem abrexex2OLD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ z E. x e. A ph
2		nfcv 2764	. . . . 5 |- F/_ y A
3		nfs1v 2437	. . . . 5 |- F/ y [ z / y ] ph
4	2, 3	nfrex 3007	. . . 4 |- F/ y E. x e. A [ z / y ] ph
5		sbequ12 2111	. . . . 5 |- ( y = z -> ( ph <-> [ z / y ] ph ) )
6	5	rexbidv 3052	. . . 4 |- ( y = z -> ( E. x e. A ph <-> E. x e. A [ z / y ] ph ) )
7	1, 4, 6	cbvab 2746	. . 3 |- { y | E. x e. A ph } = { z | E. x e. A [ z / y ] ph }
8		df-clab 2609	. . . . 5 |- ( z e. { y | ph } <-> [ z / y ] ph )
9	8	rexbii 3041	. . . 4 |- ( E. x e. A z e. { y | ph } <-> E. x e. A [ z / y ] ph )
10	9	abbii 2739	. . 3 |- { z | E. x e. A z e. { y | ph } } = { z | E. x e. A [ z / y ] ph }
11	7, 10	eqtr4i 2647	. 2 |- { y | E. x e. A ph } = { z | E. x e. A z e. { y | ph } }
12		df-iun 4522	. . 3 |- U_ x e. A { y | ph } = { z | E. x e. A z e. { y | ph } }
13		abrexex2OLD.1	. . . 4 |- A e. _V
14		abrexex2OLD.2	. . . 4 |- { y | ph } e. _V
15	13, 14	iunex 7147	. . 3 |- U_ x e. A { y | ph } e. _V
16	12, 15	eqeltrri 2698	. 2 |- { z | E. x e. A z e. { y | ph } } e. _V
17	11, 16	eqeltri 2697	1 |- { y | E. x e. A ph } e. _V

Syntax hints:   [wsb 1880   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)