Theorem ralsnsg 3659

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Assertion

Ref	Expression
ralsnsg	|- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Distinct variable group:   x, A

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

Proof of Theorem ralsnsg

Step	Hyp	Ref	Expression
1		df-ral 2480	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
2		velsn 3639	. . . . 5 |- ( x e. { A } <-> x = A )
3	2	imbi1i 238	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
4	3	albii 1484	. . 3 |- ( A. x ( x e. { A } -> ph ) <-> A. x ( x = A -> ph ) )
5	1, 4	bitri 184	. 2 |- ( A. x e. { A } ph <-> A. x ( x = A -> ph ) )
6		sbc6g 3014	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
7	5, 6	bitr4id 199	1 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2167   A.wral 2475   [.wsbc 2989   {csn 3622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-sbc 2990  df-sn 3628

This theorem is referenced by:  ixpsnval  6760  ac6sfi  6959  rexfiuz  11154  prmind2  12288