Theorem sbcsng 2753

Description: Substitution expressed in terms of quantification over a singleton.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem sbcsng

Step	Hyp	Ref	Expression
1		sbc6g 1955	. 2 |- (A e. B -> ([A / x]ph <-> A.x(x = A -> ph)))
2		df-ral 1649	. . 3 |- (A.x e. {A}ph <-> A.x(x e. {A} -> ph))
3		elsn 2421	. . . . 5 |- (x e. {A} <-> x = A)
4	3	imbi1i 186	. . . 4 |- ((x e. {A} -> ph) <-> (x = A -> ph))
5	4	albii 999	. . 3 |- (A.x(x e. {A} -> ph) <-> A.x(x = A -> ph))
6	2, 5	bitr2 174	. 2 |- (A.x(x = A -> ph) <-> A.x e. {A}ph)
7	1, 6	syl6bb 536	1 |- (A e. B -> ([A / x]ph <-> A.x e. {A}ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  [wsbc 1170  A.wral 1645  {csn 2409

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-sbc 1942  df-sn 2412

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