Theorem sbcoreleleq 38745

Description: Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 39095. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcoreleleq	|- ( A e. V -> ( [. A / y ]. ( x e. y \/ y e. x \/ x = y ) <-> ( x e. A \/ A e. x \/ x = A ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   V( x, y)

Proof of Theorem sbcoreleleq

Step	Hyp	Ref	Expression
1		sbcel2gv 3496	. . 3 |- ( A e. V -> ( [. A / y ]. x e. y <-> x e. A ) )
2		sbcel1v 3495	. . . 4 |- ( [. A / y ]. y e. x <-> A e. x )
3	2	a1i 11	. . 3 |- ( A e. V -> ( [. A / y ]. y e. x <-> A e. x ) )
4		eqsbc3r 3492	. . 3 |- ( A e. V -> ( [. A / y ]. x = y <-> x = A ) )
5		3orbi123 38717	. . . 4 |- ( ( ( [. A / y ]. x e. y <-> x e. A ) /\ ( [. A / y ]. y e. x <-> A e. x ) /\ ( [. A / y ]. x = y <-> x = A ) ) -> ( ( [. A / y ]. x e. y \/ [. A / y ]. y e. x \/ [. A / y ]. x = y ) <-> ( x e. A \/ A e. x \/ x = A ) ) )
6	5	3impexpbicomi 38686	. . 3 |- ( ( [. A / y ]. x e. y <-> x e. A ) -> ( ( [. A / y ]. y e. x <-> A e. x ) -> ( ( [. A / y ]. x = y <-> x = A ) -> ( ( x e. A \/ A e. x \/ x = A ) <-> ( [. A / y ]. x e. y \/ [. A / y ]. y e. x \/ [. A / y ]. x = y ) ) ) ) )
7	1, 3, 4, 6	syl3c 66	. 2 |- ( A e. V -> ( ( x e. A \/ A e. x \/ x = A ) <-> ( [. A / y ]. x e. y \/ [. A / y ]. y e. x \/ [. A / y ]. x = y ) ) )
8		sbc3or 38738	. 2 |- ( [. A / y ]. ( x e. y \/ y e. x \/ x = y ) <-> ( [. A / y ]. x e. y \/ [. A / y ]. y e. x \/ [. A / y ]. x = y ) )
9	7, 8	syl6rbbr 279	1 |- ( A e. V -> ( [. A / y ]. ( x e. y \/ y e. x \/ x = y ) <-> ( x e. A \/ A e. x \/ x = A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ w3o 1036   = wceq 1483   e. wcel 1990   [.wsbc 3435

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-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  tratrb  38746  tratrbVD  39097