Theorem pm13.192

Description: Theorem *13.192 in WhiteheadRussell p. 179. (Contributed by Andrew Salmon, 3-Jun-2011) (Revised by NM, 4-Jan-2017)

	Ref	Expression
Assertion	pm13.192	|- ( E. y ( A. x ( x = A <-> x = y ) /\ ph ) <-> [. A / y ]. ph )

Proof

Step	Hyp	Ref	Expression
1		biimpr	|- ( ( x = A <-> x = y ) -> ( x = y -> x = A ) )
2	1	alimi	|- ( A. x ( x = A <-> x = y ) -> A. x ( x = y -> x = A ) )
3		eqeq1	|- ( x = y -> ( x = A <-> y = A ) )
4	3	equsalvw	|- ( A. x ( x = y -> x = A ) <-> y = A )
5	2 4	sylib	|- ( A. x ( x = A <-> x = y ) -> y = A )
6		eqeq2	|- ( A = y -> ( x = A <-> x = y ) )
7	6	eqcoms	|- ( y = A -> ( x = A <-> x = y ) )
8	7	alrimiv	|- ( y = A -> A. x ( x = A <-> x = y ) )
9	5 8	impbii	|- ( A. x ( x = A <-> x = y ) <-> y = A )
10	9	anbi1i	|- ( ( A. x ( x = A <-> x = y ) /\ ph ) <-> ( y = A /\ ph ) )
11	10	exbii	|- ( E. y ( A. x ( x = A <-> x = y ) /\ ph ) <-> E. y ( y = A /\ ph ) )
12		sbc5	|- ( [. A / y ]. ph <-> E. y ( y = A /\ ph ) )
13	11 12	bitr4i	|- ( E. y ( A. x ( x = A <-> x = y ) /\ ph ) <-> [. A / y ]. ph )