Theorem rgenm 3517

Description: Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)

Hypothesis

Ref	Expression
rgenm.1	|- ( ( E. x x e. A /\ x e. A ) -> ph )

Assertion

Ref	Expression
rgenm	|- A. x e. A ph

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rgenm

Step	Hyp	Ref	Expression
1		nfe1 1489	. . . . 5 |- F/ x E. x x e. A
2		rgenm.1	. . . . . 6 |- ( ( E. x x e. A /\ x e. A ) -> ph )
3	2	ex 114	. . . . 5 |- ( E. x x e. A -> ( x e. A -> ph ) )
4	1, 3	alrimi 1515	. . . 4 |- ( E. x x e. A -> A. x ( x e. A -> ph ) )
5		19.38 1669	. . . 4 |- ( ( E. x x e. A -> A. x ( x e. A -> ph ) ) -> A. x ( x e. A -> ( x e. A -> ph ) ) )
6	4, 5	ax-mp 5	. . 3 |- A. x ( x e. A -> ( x e. A -> ph ) )
7		pm5.4 248	. . . 4 |- ( ( x e. A -> ( x e. A -> ph ) ) <-> ( x e. A -> ph ) )
8	7	albii 1463	. . 3 |- ( A. x ( x e. A -> ( x e. A -> ph ) ) <-> A. x ( x e. A -> ph ) )
9	6, 8	mpbi 144	. 2 |- A. x ( x e. A -> ph )
10		df-ral 2453	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
11	9, 10	mpbir 145	1 |- A. x e. A ph

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   E.wex 1485   e. wcel 2141   A.wral 2448

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-4 1503  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453

This theorem is referenced by: (None)