Theorem bnj610 30817

Description: Pass from equality ( x = A) to substitution ( [. A / x ].) without the distinct variable restriction ($d A x). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj610.1	|- A e. _V
bnj610.2	|- ( x = A -> ( ph <-> ps ) )
bnj610.3	|- ( x = y -> ( ph <-> ps' ) )
bnj610.4	|- ( y = A -> ( ps' <-> ps ) )

Assertion

Ref	Expression
bnj610	|- ( [. A / x ]. ph <-> ps )

Distinct variable groups:   y, A   ph, y   ps, y   x, ps'   x, y

Allowed substitution hints:   ph( x)   ps( x)   A( x)   ps'( y)

Proof of Theorem bnj610

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- y e. _V
2		bnj610.3	. . . 4 |- ( x = y -> ( ph <-> ps' ) )
3	1, 2	sbcie 3470	. . 3 |- ( [. y / x ]. ph <-> ps' )
4	3	sbcbii 3491	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / y ]. ps' )
5		sbcco 3458	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )
6		bnj610.1	. . 3 |- A e. _V
7		bnj610.4	. . 3 |- ( y = A -> ( ps' <-> ps ) )
8	6, 7	sbcie 3470	. 2 |- ( [. A / y ]. ps' <-> ps )
9	4, 5, 8	3bitr3i 290	1 |- ( [. A / x ]. ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  bnj611  30988  bnj1000  31011