Theorem pm13.192 38611

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

Assertion

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

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem pm13.192

Step	Hyp	Ref	Expression
1		biimpr 210	. . . . . . 7 |- ( ( x = A <-> x = y ) -> ( x = y -> x = A ) )
2	1	alimi 1739	. . . . . 6 |- ( A. x ( x = A <-> x = y ) -> A. x ( x = y -> x = A ) )
3		eqeq1 2626	. . . . . . 7 |- ( x = y -> ( x = A <-> y = A ) )
4	3	equsalvw 1931	. . . . . 6 |- ( A. x ( x = y -> x = A ) <-> y = A )
5	2, 4	sylib 208	. . . . 5 |- ( A. x ( x = A <-> x = y ) -> y = A )
6		eqeq2 2633	. . . . . . 7 |- ( A = y -> ( x = A <-> x = y ) )
7	6	eqcoms 2630	. . . . . 6 |- ( y = A -> ( x = A <-> x = y ) )
8	7	alrimiv 1855	. . . . 5 |- ( y = A -> A. x ( x = A <-> x = y ) )
9	5, 8	impbii 199	. . . 4 |- ( A. x ( x = A <-> x = y ) <-> y = A )
10	9	anbi1i 731	. . 3 |- ( ( A. x ( x = A <-> x = y ) /\ ph ) <-> ( y = A /\ ph ) )
11	10	exbii 1774	. 2 |- ( E. y ( A. x ( x = A <-> x = y ) /\ ph ) <-> E. y ( y = A /\ ph ) )
12		sbc5 3460	. 2 |- ( [. A / y ]. ph <-> E. y ( y = A /\ ph ) )
13	11, 12	bitr4i 267	1 |- ( E. y ( A. x ( x = A <-> x = y ) /\ ph ) <-> [. A / y ]. ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   [.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-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

This theorem is referenced by: (None)