Theorem sbc6g 2472

Description: An equivalence for class substitution. (The proof was shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
sbc6g	|- (A e. B -> ([A / x]ph <-> A.x(x = A -> ph)))

Distinct variable group:   x,A

Proof of Theorem sbc6g

Step	Hyp	Ref	Expression
1		sbc5g 2470	. 2 |- (A e. B -> ([A / x]ph <-> E.x(x = A /\ ph)))
2		hbe1 1363	. . 3 |- (E.x(x = A /\ ph) -> A.xE.x(x = A /\ ph))
3		ceqex 2391	. . 3 |- (x = A -> (ph <-> E.x(x = A /\ ph)))
4	2, 3	ceqsalg 2314	. 2 |- (A e. B -> (A.x(x = A -> ph) <-> E.x(x = A /\ ph)))
5	1, 4	bitr4d 590	1 |- (A e. B -> ([A / x]ph <-> A.x(x = A -> ph)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534

This theorem is referenced by:  sbc6 2475  sbciegft 2482  sbcralt 2527  sbcralgf 2529  ralprg 3078  ralsng 3085  sbcsngOLD 3087  fz1sbc 7696  pm14.122a 16386

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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