Theorem eqsbc3rVD 37230

Description: Virtual deduction proof of eqsbc3r 3323. (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 groups:   x, C   x, A

Allowed substitution hint:   B( x)

Proof of Theorem eqsbc3rVD

Step	Hyp	Ref	Expression
1		idn1 36938	. . . . . . 7 |- (. A e. B ->. A e. B ).
2		eqsbc3 3306	. . . . . . 7 |- ( A e. B -> ( [. A / x ]. x = C <-> A = C ) )
3	1, 2	e1a 37000	. . . . . 6 |- (. A e. B ->. ( [. A / x ]. x = C <-> A = C ) ).
4		eqcom 2457	. . . . . . . . 9 |- ( C = x <-> x = C )
5	4	sbcbiiOLD 36886	. . . . . . . 8 |- ( A e. B -> ( [. A / x ]. C = x <-> [. A / x ]. x = C ) )
6	1, 5	e1a 37000	. . . . . . 7 |- (. A e. B ->. ( [. A / x ]. C = x <-> [. A / x ]. x = C ) ).
7		idn2 36986	. . . . . . 7 |- (. A e. B ,. [. A / x ]. C = x ->. [. A / x ]. C = x ).
8		biimp 197	. . . . . . 7 |- ( ( [. A / x ]. C = x <-> [. A / x ]. x = C ) -> ( [. A / x ]. C = x -> [. A / x ]. x = C ) )
9	6, 7, 8	e12 37105	. . . . . 6 |- (. A e. B ,. [. A / x ]. C = x ->. [. A / x ]. x = C ).
10		biimp 197	. . . . . 6 |- ( ( [. A / x ]. x = C <-> A = C ) -> ( [. A / x ]. x = C -> A = C ) )
11	3, 9, 10	e12 37105	. . . . 5 |- (. A e. B ,. [. A / x ]. C = x ->. A = C ).
12		eqcom 2457	. . . . 5 |- ( A = C <-> C = A )
13	11, 12	e2bi 37005	. . . 4 |- (. A e. B ,. [. A / x ]. C = x ->. C = A ).
14	13	in2 36978	. . 3 |- (. A e. B ->. ( [. A / x ]. C = x -> C = A ) ).
15		idn2 36986	. . . . . . 7 |- (. A e. B ,. C = A ->. C = A ).
16	15, 12	e2bir 37006	. . . . . 6 |- (. A e. B ,. C = A ->. A = C ).
17		biimpr 202	. . . . . 6 |- ( ( [. A / x ]. x = C <-> A = C ) -> ( A = C -> [. A / x ]. x = C ) )
18	3, 16, 17	e12 37105	. . . . 5 |- (. A e. B ,. C = A ->. [. A / x ]. x = C ).
19		biimpr 202	. . . . 5 |- ( ( [. A / x ]. C = x <-> [. A / x ]. x = C ) -> ( [. A / x ]. x = C -> [. A / x ]. C = x ) )
20	6, 18, 19	e12 37105	. . . 4 |- (. A e. B ,. C = A ->. [. A / x ]. C = x ).
21	20	in2 36978	. . 3 |- (. A e. B ->. ( C = A -> [. A / x ]. C = x ) ).
22		impbi 190	. . 3 |- ( ( [. A / x ]. C = x -> C = A ) -> ( ( C = A -> [. A / x ]. C = x ) -> ( [. A / x ]. C = x <-> C = A ) ) )
23	14, 21, 22	e11 37061	. 2 |- (. A e. B ->. ( [. A / x ]. C = x <-> C = A ) ).
24	23	in1 36935	1 |- ( A e. B -> ( [. A / x ]. C = x <-> C = A ) )

Syntax hints:   -> wi 4   <-> wb 188   = wceq 1443   e. wcel 1886   [.wsbc 3266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3046  df-sbc 3267  df-vd1 36934  df-vd2 36942

This theorem is referenced by: (None)