Theorem bnj89 30787

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj89.1	|- Z e. _V

Assertion

Ref	Expression
bnj89	|- ( [. Z / y ]. E! x ph <-> E! x [. Z / y ]. ph )

Distinct variable groups:   x, Z   x, y

Allowed substitution hints:   ph( x, y)   Z( y)

Proof of Theorem bnj89

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex2 3486	. . 3 |- ( [. Z / y ]. E. w A. x ( ph <-> x = w ) <-> E. w [. Z / y ]. A. x ( ph <-> x = w ) )
2		sbcal 3485	. . . 4 |- ( [. Z / y ]. A. x ( ph <-> x = w ) <-> A. x [. Z / y ]. ( ph <-> x = w ) )
3	2	exbii 1774	. . 3 |- ( E. w [. Z / y ]. A. x ( ph <-> x = w ) <-> E. w A. x [. Z / y ]. ( ph <-> x = w ) )
4		bnj89.1	. . . . . . 7 |- Z e. _V
5		sbcbig 3480	. . . . . . 7 |- ( Z e. _V -> ( [. Z / y ]. ( ph <-> x = w ) <-> ( [. Z / y ]. ph <-> [. Z / y ]. x = w ) ) )
6	4, 5	ax-mp 5	. . . . . 6 |- ( [. Z / y ]. ( ph <-> x = w ) <-> ( [. Z / y ]. ph <-> [. Z / y ]. x = w ) )
7		sbcg 3503	. . . . . . . 8 |- ( Z e. _V -> ( [. Z / y ]. x = w <-> x = w ) )
8	4, 7	ax-mp 5	. . . . . . 7 |- ( [. Z / y ]. x = w <-> x = w )
9	8	bibi2i 327	. . . . . 6 |- ( ( [. Z / y ]. ph <-> [. Z / y ]. x = w ) <-> ( [. Z / y ]. ph <-> x = w ) )
10	6, 9	bitri 264	. . . . 5 |- ( [. Z / y ]. ( ph <-> x = w ) <-> ( [. Z / y ]. ph <-> x = w ) )
11	10	albii 1747	. . . 4 |- ( A. x [. Z / y ]. ( ph <-> x = w ) <-> A. x ( [. Z / y ]. ph <-> x = w ) )
12	11	exbii 1774	. . 3 |- ( E. w A. x [. Z / y ]. ( ph <-> x = w ) <-> E. w A. x ( [. Z / y ]. ph <-> x = w ) )
13	1, 3, 12	3bitri 286	. 2 |- ( [. Z / y ]. E. w A. x ( ph <-> x = w ) <-> E. w A. x ( [. Z / y ]. ph <-> x = w ) )
14		df-eu 2474	. . 3 |- ( E! x ph <-> E. w A. x ( ph <-> x = w ) )
15	14	sbcbii 3491	. 2 |- ( [. Z / y ]. E! x ph <-> [. Z / y ]. E. w A. x ( ph <-> x = w ) )
16		df-eu 2474	. 2 |- ( E! x [. Z / y ]. ph <-> E. w A. x ( [. Z / y ]. ph <-> x = w ) )
17	13, 15, 16	3bitr4i 292	1 |- ( [. Z / y ]. E! x ph <-> E! x [. Z / y ]. ph )

Syntax hints:   <-> wb 196   A.wal 1481   E.wex 1704   e. wcel 1990   E!weu 2470   _Vcvv 3200   [.wsbc 3435

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  bnj130  30944  bnj207  30951