Theorem prtlem5 34145

Description: Lemma for prter1 34164, prter2 34166, prter3 34167 and prtex 34165. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Assertion

Ref	Expression
prtlem5	|- ( [ s / v ] [ r / u ] E. x e. A ( u e. x /\ v e. x ) <-> E. x e. A ( r e. x /\ s e. x ) )

Distinct variable groups:   v, u, x, r   u, s, v, x   u, A, v, x

Allowed substitution hints:   A( s, r)

Proof of Theorem prtlem5

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ v E. x e. A ( r e. x /\ s e. x )
2		elequ1 1997	. . . . 5 |- ( u = r -> ( u e. x <-> r e. x ) )
3		elequ1 1997	. . . . 5 |- ( v = s -> ( v e. x <-> s e. x ) )
4	2, 3	bi2anan9r 918	. . . 4 |- ( ( v = s /\ u = r ) -> ( ( u e. x /\ v e. x ) <-> ( r e. x /\ s e. x ) ) )
5	4	rexbidv 3052	. . 3 |- ( ( v = s /\ u = r ) -> ( E. x e. A ( u e. x /\ v e. x ) <-> E. x e. A ( r e. x /\ s e. x ) ) )
6	5	sbiedv 2410	. 2 |- ( v = s -> ( [ r / u ] E. x e. A ( u e. x /\ v e. x ) <-> E. x e. A ( r e. x /\ s e. x ) ) )
7	1, 6	sbie 2408	1 |- ( [ s / v ] [ r / u ] E. x e. A ( u e. x /\ v e. x ) <-> E. x e. A ( r e. x /\ s e. x ) )

Syntax hints:   <-> wb 196   /\ wa 384   [wsb 1880   E.wrex 2913

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-8 1992  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881  df-rex 2918

This theorem is referenced by: (None)