Theorem dfon2lem1 31688

Description: Lemma for dfon2 31697. (Contributed by Scott Fenton, 28-Feb-2011.)

Assertion

Ref	Expression
dfon2lem1	|- Tr U. { x | ( ph /\ Tr x /\ ps ) }

Proof of Theorem dfon2lem1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		truni 4767	. 2 |- ( A. y e. { x | ( ph /\ Tr x /\ ps ) } Tr y -> Tr U. { x | ( ph /\ Tr x /\ ps ) } )
2		nfsbc1v 3455	. . . . 5 |- F/ x [. y / x ]. ph
3		nfv 1843	. . . . 5 |- F/ x Tr y
4		nfsbc1v 3455	. . . . 5 |- F/ x [. y / x ]. ps
5	2, 3, 4	nf3an 1831	. . . 4 |- F/ x ( [. y / x ]. ph /\ Tr y /\ [. y / x ]. ps )
6		vex 3203	. . . 4 |- y e. _V
7		sbceq1a 3446	. . . . 5 |- ( x = y -> ( ph <-> [. y / x ]. ph ) )
8		treq 4758	. . . . 5 |- ( x = y -> ( Tr x <-> Tr y ) )
9		sbceq1a 3446	. . . . 5 |- ( x = y -> ( ps <-> [. y / x ]. ps ) )
10	7, 8, 9	3anbi123d 1399	. . . 4 |- ( x = y -> ( ( ph /\ Tr x /\ ps ) <-> ( [. y / x ]. ph /\ Tr y /\ [. y / x ]. ps ) ) )
11	5, 6, 10	elabf 3349	. . 3 |- ( y e. { x | ( ph /\ Tr x /\ ps ) } <-> ( [. y / x ]. ph /\ Tr y /\ [. y / x ]. ps ) )
12	11	simp2bi 1077	. 2 |- ( y e. { x | ( ph /\ Tr x /\ ps ) } -> Tr y )
13	1, 12	mprg 2926	1 |- Tr U. { x | ( ph /\ Tr x /\ ps ) }

Syntax hints:   /\ w3a 1037   e. wcel 1990   {cab 2608   [.wsbc 3435   U.cuni 4436   Tr wtr 4752

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-3an 1039  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-rex 2918  df-v 3202  df-sbc 3436  df-in 3581  df-ss 3588  df-uni 4437  df-iun 4522  df-tr 4753

This theorem is referenced by:  dfon2lem3  31690  dfon2lem7  31694