Theorem bj-eu3f 32829

Description: Version of eu3v 2498 where the dv condition is replaced with a non-freeness hypothesis. This is a "backup" of a theorem that used to be in the main part with label "eu3" and was deprecated in favor of eu3v 2498. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)

Hypothesis

Ref	Expression
bj-eu3f.1	|- F/ y ph

Assertion

Ref	Expression
bj-eu3f	|- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-eu3f

Step	Hyp	Ref	Expression
1		eu5 2496	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		bj-eu3f.1	. . . 4 |- F/ y ph
3	2	mo2 2479	. . 3 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
4	3	anbi2i 730	. 2 |- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
5	1, 4	bitri 264	1 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   E!weu 2470   E*wmo 2471

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)