Theorem rmoanim 41179

Description: Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2529. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Hypothesis

Ref	Expression
rmoanim.1	|- F/ x ph

Assertion

Ref	Expression
rmoanim	|- ( E* x e. A ( ph /\ ps ) <-> ( ph -> E* x e. A ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem rmoanim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		impexp 462	. . . . 5 |- ( ( ( ph /\ ps ) -> x = y ) <-> ( ph -> ( ps -> x = y ) ) )
2	1	ralbii 2980	. . . 4 |- ( A. x e. A ( ( ph /\ ps ) -> x = y ) <-> A. x e. A ( ph -> ( ps -> x = y ) ) )
3		rmoanim.1	. . . . 5 |- F/ x ph
4	3	r19.21 2956	. . . 4 |- ( A. x e. A ( ph -> ( ps -> x = y ) ) <-> ( ph -> A. x e. A ( ps -> x = y ) ) )
5	2, 4	bitri 264	. . 3 |- ( A. x e. A ( ( ph /\ ps ) -> x = y ) <-> ( ph -> A. x e. A ( ps -> x = y ) ) )
6	5	exbii 1774	. 2 |- ( E. y A. x e. A ( ( ph /\ ps ) -> x = y ) <-> E. y ( ph -> A. x e. A ( ps -> x = y ) ) )
7		nfv 1843	. . 3 |- F/ y ( ph /\ ps )
8	7	rmo2 3526	. 2 |- ( E* x e. A ( ph /\ ps ) <-> E. y A. x e. A ( ( ph /\ ps ) -> x = y ) )
9		nfv 1843	. . . . 5 |- F/ y ps
10	9	rmo2 3526	. . . 4 |- ( E* x e. A ps <-> E. y A. x e. A ( ps -> x = y ) )
11	10	imbi2i 326	. . 3 |- ( ( ph -> E* x e. A ps ) <-> ( ph -> E. y A. x e. A ( ps -> x = y ) ) )
12		19.37v 1910	. . 3 |- ( E. y ( ph -> A. x e. A ( ps -> x = y ) ) <-> ( ph -> E. y A. x e. A ( ps -> x = y ) ) )
13	11, 12	bitr4i 267	. 2 |- ( ( ph -> E* x e. A ps ) <-> E. y ( ph -> A. x e. A ( ps -> x = y ) ) )
14	6, 8, 13	3bitr4i 292	1 |- ( E* x e. A ( ph /\ ps ) <-> ( ph -> E* x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708   A.wral 2912   E*wrmo 2915

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-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-ral 2917  df-rmo 2920

This theorem is referenced by:  2reu1  41186