Theorem raldifeq 4059

Description: Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)

Hypotheses

Ref	Expression
raldifeq.1	|- ( ph -> A C_ B )
raldifeq.2	|- ( ph -> A. x e. ( B \ A ) ps )

Assertion

Ref	Expression
raldifeq	|- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem raldifeq

Step	Hyp	Ref	Expression
1		raldifeq.2	. . . 4 |- ( ph -> A. x e. ( B \ A ) ps )
2	1	biantrud 528	. . 3 |- ( ph -> ( A. x e. A ps <-> ( A. x e. A ps /\ A. x e. ( B \ A ) ps ) ) )
3		ralunb 3794	. . 3 |- ( A. x e. ( A u. ( B \ A ) ) ps <-> ( A. x e. A ps /\ A. x e. ( B \ A ) ps ) )
4	2, 3	syl6bbr 278	. 2 |- ( ph -> ( A. x e. A ps <-> A. x e. ( A u. ( B \ A ) ) ps ) )
5		raldifeq.1	. . . 4 |- ( ph -> A C_ B )
6		undif 4049	. . . 4 |- ( A C_ B <-> ( A u. ( B \ A ) ) = B )
7	5, 6	sylib 208	. . 3 |- ( ph -> ( A u. ( B \ A ) ) = B )
8	7	raleqdv 3144	. 2 |- ( ph -> ( A. x e. ( A u. ( B \ A ) ) ps <-> A. x e. B ps ) )
9	4, 8	bitrd 268	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   A.wral 2912   \ cdif 3571   u. cun 3572   C_ wss 3574

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  cantnfrescl  8573  rrxmet  23191  ntrneiel2  38384  ntrneik4w  38398