Theorem sbcsng 2748

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 1951	. 2 |- (A e. B -> ([A / x]ph <-> A.x(x = A -> ph)))
2		df-ral 1646	. . 3 |- (A.x e. {A}ph <-> A.x(x e. {A} -> ph))
3		elsn 2417	. . . . 5 |- (x e. {A} <-> x = A)
4	3	imbi1i 186	. . . 4 |- ((x e. {A} -> ph) <-> (x = A -> ph))
5	4	albii 997	. . 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 535	1 |- (A e. B -> ([A / x]ph <-> A.x e. {A}ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  [wsbc 1168  A.wral 1642  {csn 2405

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-sbc 1938  df-sn 2408

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