Theorem vtoclg1f 2789

Description: Version of vtoclgf 2788 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1499 and ax-13 2143. (Contributed by BJ, 1-May-2019.)

Hypotheses

Ref	Expression
vtoclg1f.nf	|- F/ x ps
vtoclg1f.maj	|- ( x = A -> ( ph <-> ps ) )
vtoclg1f.min	|- ph

Assertion

Ref	Expression
vtoclg1f	|- ( A e. V -> ps )

Distinct variable group:   x, A

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

Proof of Theorem vtoclg1f

Step	Hyp	Ref	Expression
1		elex 2741	. 2 |- ( A e. V -> A e. _V )
2		isset 2736	. . 3 |- ( A e. _V <-> E. x x = A )
3		vtoclg1f.nf	. . . 4 |- F/ x ps
4		vtoclg1f.min	. . . . 5 |- ph
5		vtoclg1f.maj	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
6	4, 5	mpbii 147	. . . 4 |- ( x = A -> ps )
7	3, 6	exlimi 1587	. . 3 |- ( E. x x = A -> ps )
8	2, 7	sylbi 120	. 2 |- ( A e. _V -> ps )
9	1, 8	syl 14	1 |- ( A e. V -> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   F/wnf 1453   E.wex 1485   e. wcel 2141   _Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  opeliunxp2f  6217  summodclem2a  11344  fprodsplit1f  11597