Theorem a4sbc 1945

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 1185 and ra4sbc 1997.

Assertion

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

Proof of Theorem a4sbc

Step	Hyp	Ref	Expression
1		dfsbcq 1943	. . 3 |- (y = A -> ([y / x]ph <-> [A / x]ph))
2		stdpc4 1185	. . 3 |- (A.xph -> [y / x]ph)
3	1, 2	syl5bi 208	. 2 |- (y = A -> (A.xph -> [A / x]ph))
4	3	vtocleg 1855	1 |- (A e. B -> (A.xph -> [A / x]ph))

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  [wsbc 1170

This theorem is referenced by:  sbcth 1946  sbcthdv 1947  sbcbid 1976  sbc19.20dv 1985  csbexg 2008  csbeq2d 2018  intab 2560

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  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-v 1812  df-sbc 1942

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