Theorem eqsbc3rVD 39075

Description: Virtual deduction proof of eqsbc3r 3492. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
eqsbc3rVD	|- ( A e. B -> ( [. A / x ]. C = x <-> C = A ) )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem eqsbc3rVD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . . . 7 |- (. A e. B ->. A e. B ).
2		eqsbc3 3475	. . . . . . 7 |- ( A e. B -> ( [. A / x ]. x = C <-> A = C ) )
3	1, 2	e1a 38852	. . . . . 6 |- (. A e. B ->. ( [. A / x ]. x = C <-> A = C ) ).
4		eqcom 2629	. . . . . . . . 9 |- ( C = x <-> x = C )
5	4	sbcbiiOLD 38741	. . . . . . . 8 |- ( A e. B -> ( [. A / x ]. C = x <-> [. A / x ]. x = C ) )
6	1, 5	e1a 38852	. . . . . . 7 |- (. A e. B ->. ( [. A / x ]. C = x <-> [. A / x ]. x = C ) ).
7		idn2 38838	. . . . . . 7 |- (. A e. B ,. [. A / x ]. C = x ->. [. A / x ]. C = x ).
8		biimp 205	. . . . . . 7 |- ( ( [. A / x ]. C = x <-> [. A / x ]. x = C ) -> ( [. A / x ]. C = x -> [. A / x ]. x = C ) )
9	6, 7, 8	e12 38951	. . . . . 6 |- (. A e. B ,. [. A / x ]. C = x ->. [. A / x ]. x = C ).
10		biimp 205	. . . . . 6 |- ( ( [. A / x ]. x = C <-> A = C ) -> ( [. A / x ]. x = C -> A = C ) )
11	3, 9, 10	e12 38951	. . . . 5 |- (. A e. B ,. [. A / x ]. C = x ->. A = C ).
12		eqcom 2629	. . . . 5 |- ( A = C <-> C = A )
13	11, 12	e2bi 38857	. . . 4 |- (. A e. B ,. [. A / x ]. C = x ->. C = A ).
14	13	in2 38830	. . 3 |- (. A e. B ->. ( [. A / x ]. C = x -> C = A ) ).
15		idn2 38838	. . . . . . 7 |- (. A e. B ,. C = A ->. C = A ).
16	15, 12	e2bir 38858	. . . . . 6 |- (. A e. B ,. C = A ->. A = C ).
17		biimpr 210	. . . . . 6 |- ( ( [. A / x ]. x = C <-> A = C ) -> ( A = C -> [. A / x ]. x = C ) )
18	3, 16, 17	e12 38951	. . . . 5 |- (. A e. B ,. C = A ->. [. A / x ]. x = C ).
19		biimpr 210	. . . . 5 |- ( ( [. A / x ]. C = x <-> [. A / x ]. x = C ) -> ( [. A / x ]. x = C -> [. A / x ]. C = x ) )
20	6, 18, 19	e12 38951	. . . 4 |- (. A e. B ,. C = A ->. [. A / x ]. C = x ).
21	20	in2 38830	. . 3 |- (. A e. B ->. ( C = A -> [. A / x ]. C = x ) ).
22		impbi 198	. . 3 |- ( ( [. A / x ]. C = x -> C = A ) -> ( ( C = A -> [. A / x ]. C = x ) -> ( [. A / x ]. C = x <-> C = A ) ) )
23	14, 21, 22	e11 38913	. 2 |- (. A e. B ->. ( [. A / x ]. C = x <-> C = A ) ).
24	23	in1 38787	1 |- ( A e. B -> ( [. A / x ]. C = x <-> C = A ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   [.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-v 3202  df-sbc 3436  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)