Theorem rgenm 3323

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 1385	. . . . 5 |- F/ x E. x x e. A
2		rgenm.1	. . . . . 6 |- ( ( E. x x e. A /\ x e. A ) -> ph )
3	2	ex 108	. . . . 5 |- ( E. x x e. A -> ( x e. A -> ph ) )
4	1, 3	alrimi 1415	. . . 4 |- ( E. x x e. A -> A. x ( x e. A -> ph ) )
5		19.38 1566	. . . 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 7	. . 3 |- A. x ( x e. A -> ( x e. A -> ph ) )
7		pm5.4 238	. . . 4 |- ( ( x e. A -> ( x e. A -> ph ) ) <-> ( x e. A -> ph ) )
8	7	albii 1359	. . 3 |- ( A. x ( x e. A -> ( x e. A -> ph ) ) <-> A. x ( x e. A -> ph ) )
9	6, 8	mpbi 133	. 2 |- A. x ( x e. A -> ph )
10		df-ral 2311	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
11	9, 10	mpbir 134	1 |- A. x e. A ph

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   E.wex 1381   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311

This theorem is referenced by: (None)