Theorem a4sbc 1948

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 1187 and ra4sbc 2000.

Assertion

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

Proof of Theorem a4sbc

Step	Hyp	Ref	Expression
1		dfsbcq 1946	. . 3 |- (y = A -> ([y / x]ph <-> [A / x]ph))
2		stdpc4 1187	. . 3 |- (A.xph -> [y / x]ph)
3	1, 2	syl5bi 208	. 2 |- (y = A -> (A.xph -> [A / x]ph))
4	3	vtocleg 1858	1 |- (A e. B -> (A.xph -> [A / x]ph))

Syntax hints:   -> wi 3  A.wal 956   = wceq 958   e. wcel 960  [wsbc 1172

This theorem is referenced by:  sbcth 1949  sbcthdv 1950  sbcbid 1979  sbc19.20dv 1988  csbexg 2011  csbeq2d 2021  intab 2564

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945

Copyright terms: Public domain