Theorem rexsb 41168

Description: An equivalent expression for restricted existence, analogous to exsb 2468. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Assertion

Ref	Expression
rexsb	|- ( E. x e. A ph <-> E. y e. A A. x ( x = y -> ph ) )

Distinct variable groups:   x, y, A   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem rexsb

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ y ph
2		nfa1 2028	. 2 |- F/ x A. x ( x = y -> ph )
3		ax12v 2048	. . 3 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
4		sp 2053	. . . 4 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
5	4	com12 32	. . 3 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
6	3, 5	impbid 202	. 2 |- ( x = y -> ( ph <-> A. x ( x = y -> ph ) ) )
7	1, 2, 6	cbvrex 3168	1 |- ( E. x e. A ph <-> E. y e. A A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   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-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-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918

This theorem is referenced by:  rexrsb  41169  2rexsb  41170