Theorem a4sbc 1935

Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1181 and ra4sbc 1987.

Assertion

Ref	Expression
a4sbc	|- (A e. B -> (A.xph -> [A / x]ph))

Proof of Theorem a4sbc

Step	Hyp	Ref	Expression
1		dfsbcq 1933	. . 3 |- (y = A -> ([y / x]ph <-> [A / x]ph))
2		stdpc4 1181	. . 3 |- (A.xph -> [y / x]ph)
3	1, 2	syl5bi 208	. 2 |- (y = A -> (A.xph -> [A / x]ph))
4	3	vtocleg 1846	1 |- (A e. B -> (A.xph -> [A / x]ph))

Syntax hints:   -> wi 3  A.wal 951   = wceq 953   e. wcel 955  [wsbc 1166

This theorem is referenced by:  sbcth 1936  sbcthdv 1937  sbcbid 1966  sbc19.20dv 1975  csbexg 1998  csbeq2d 2008  intab 2550

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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