Theorem bnj538OLD 30810

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Obsolete version of bnj538 30809 as of 30-Mar-2020. (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj538OLD.1	|- A e. _V

Assertion

Ref	Expression
bnj538OLD	|- ( [. A / y ]. A. x e. B ph <-> A. x e. B [. A / y ]. ph )

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

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

Proof of Theorem bnj538OLD

Step	Hyp	Ref	Expression
1		df-ral 2917	. . 3 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
2	1	sbcbii 3491	. 2 |- ( [. A / y ]. A. x e. B ph <-> [. A / y ]. A. x ( x e. B -> ph ) )
3		bnj538OLD.1	. . . . . 6 |- A e. _V
4		sbcimg 3477	. . . . . 6 |- ( A e. _V -> ( [. A / y ]. ( x e. B -> ph ) <-> ( [. A / y ]. x e. B -> [. A / y ]. ph ) ) )
5	3, 4	ax-mp 5	. . . . 5 |- ( [. A / y ]. ( x e. B -> ph ) <-> ( [. A / y ]. x e. B -> [. A / y ]. ph ) )
6	3	bnj525 30807	. . . . . 6 |- ( [. A / y ]. x e. B <-> x e. B )
7	6	imbi1i 339	. . . . 5 |- ( ( [. A / y ]. x e. B -> [. A / y ]. ph ) <-> ( x e. B -> [. A / y ]. ph ) )
8	5, 7	bitri 264	. . . 4 |- ( [. A / y ]. ( x e. B -> ph ) <-> ( x e. B -> [. A / y ]. ph ) )
9	8	albii 1747	. . 3 |- ( A. x [. A / y ]. ( x e. B -> ph ) <-> A. x ( x e. B -> [. A / y ]. ph ) )
10		sbcal 3485	. . 3 |- ( [. A / y ]. A. x ( x e. B -> ph ) <-> A. x [. A / y ]. ( x e. B -> ph ) )
11		df-ral 2917	. . 3 |- ( A. x e. B [. A / y ]. ph <-> A. x ( x e. B -> [. A / y ]. ph ) )
12	9, 10, 11	3bitr4i 292	. 2 |- ( [. A / y ]. A. x ( x e. B -> ph ) <-> A. x e. B [. A / y ]. ph )
13	2, 12	bitri 264	1 |- ( [. A / y ]. A. x e. B ph <-> A. x e. B [. A / y ]. ph )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   A.wral 2912   _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-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)