Theorem ralbinrald 41199

Description: Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)

Hypotheses

Ref	Expression
ralbinrald.1	|- ( ph -> X e. A )
ralbinrald.2	|- ( x e. A -> x = X )
ralbinrald.3	|- ( x = X -> ( ps <-> th ) )

Assertion

Ref	Expression
ralbinrald	|- ( ph -> ( A. x e. A ps <-> th ) )

Distinct variable groups:   x, X   x, A   ph, x   th, x

Allowed substitution hint:   ps( x)

Proof of Theorem ralbinrald

Step	Hyp	Ref	Expression
1		ralbinrald.1	. . 3 |- ( ph -> X e. A )
2		ralbinrald.3	. . . 4 |- ( x = X -> ( ps <-> th ) )
3	2	adantl 482	. . 3 |- ( ( ph /\ x = X ) -> ( ps <-> th ) )
4	1, 3	rspcdv 3312	. 2 |- ( ph -> ( A. x e. A ps -> th ) )
5		ralbinrald.2	. . . . . 6 |- ( x e. A -> x = X )
6	2	bicomd 213	. . . . . 6 |- ( x = X -> ( th <-> ps ) )
7	5, 6	syl 17	. . . . 5 |- ( x e. A -> ( th <-> ps ) )
8	7	adantl 482	. . . 4 |- ( ( ph /\ x e. A ) -> ( th <-> ps ) )
9	8	biimpd 219	. . 3 |- ( ( ph /\ x e. A ) -> ( th -> ps ) )
10	9	ralrimdva 2969	. 2 |- ( ph -> ( th -> A. x e. A ps ) )
11	4, 10	impbid 202	1 |- ( ph -> ( A. x e. A ps <-> th ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by:  dfdfat2  41211