Theorem eqsbc3rVD 32595

Description: Virtual deduction proof of eqsbc3r 3386. (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 32306	. . . . . . 7 |- (. A e. B ->. A e. B ).
2		eqsbc3 3364	. . . . . . 7 |- ( A e. B -> ( [. A / x ]. x = C <-> A = C ) )
3	1, 2	e1a 32368	. . . . . 6 |- (. A e. B ->. ( [. A / x ]. x = C <-> A = C ) ).
4		eqcom 2469	. . . . . . . . 9 |- ( C = x <-> x = C )
5	4	sbcbiiOLD 3385	. . . . . . . 8 |- ( A e. B -> ( [. A / x ]. C = x <-> [. A / x ]. x = C ) )
6	1, 5	e1a 32368	. . . . . . 7 |- (. A e. B ->. ( [. A / x ]. C = x <-> [. A / x ]. x = C ) ).
7		idn2 32354	. . . . . . 7 |- (. A e. B ,. [. A / x ]. C = x ->. [. A / x ]. C = x ).
8		bi1 186	. . . . . . 7 |- ( ( [. A / x ]. C = x <-> [. A / x ]. x = C ) -> ( [. A / x ]. C = x -> [. A / x ]. x = C ) )
9	6, 7, 8	e12 32476	. . . . . 6 |- (. A e. B ,. [. A / x ]. C = x ->. [. A / x ]. x = C ).
10		bi1 186	. . . . . 6 |- ( ( [. A / x ]. x = C <-> A = C ) -> ( [. A / x ]. x = C -> A = C ) )
11	3, 9, 10	e12 32476	. . . . 5 |- (. A e. B ,. [. A / x ]. C = x ->. A = C ).
12		eqcom 2469	. . . . 5 |- ( A = C <-> C = A )
13	11, 12	e2bi 32373	. . . 4 |- (. A e. B ,. [. A / x ]. C = x ->. C = A ).
14	13	in2 32346	. . 3 |- (. A e. B ->. ( [. A / x ]. C = x -> C = A ) ).
15		idn2 32354	. . . . . . 7 |- (. A e. B ,. C = A ->. C = A ).
16	15, 12	e2bir 32374	. . . . . 6 |- (. A e. B ,. C = A ->. A = C ).
17		bi2 198	. . . . . 6 |- ( ( [. A / x ]. x = C <-> A = C ) -> ( A = C -> [. A / x ]. x = C ) )
18	3, 16, 17	e12 32476	. . . . 5 |- (. A e. B ,. C = A ->. [. A / x ]. x = C ).
19		bi2 198	. . . . 5 |- ( ( [. A / x ]. C = x <-> [. A / x ]. x = C ) -> ( [. A / x ]. x = C -> [. A / x ]. C = x ) )
20	6, 18, 19	e12 32476	. . . 4 |- (. A e. B ,. C = A ->. [. A / x ]. C = x ).
21	20	in2 32346	. . 3 |- (. A e. B ->. ( C = A -> [. A / x ]. C = x ) ).
22		bi3 187	. . 3 |- ( ( [. A / x ]. C = x -> C = A ) -> ( ( C = A -> [. A / x ]. C = x ) -> ( [. A / x ]. C = x <-> C = A ) ) )
23	14, 21, 22	e11 32429	. 2 |- (. A e. B ->. ( [. A / x ]. C = x <-> C = A ) ).
24	23	in1 32303	1 |- ( A e. B -> ( [. A / x ]. C = x <-> C = A ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1374   e. wcel 1762   [.wsbc 3324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-v 3108  df-sbc 3325  df-vd1 32302  df-vd2 32310

This theorem is referenced by: (None)