Theorem stdpc4 1185

Description: The specialization axiom of standard predicate calculus. It states that if a statement ph holds for all x, then it also holds for the specific case of y (properly) substituted for x. Translated to traditional notation, it can be read: "A.xph(x) -> ph(y), provided that y is free for x in ph(x)." Axiom 4 of [Mendelson] p. 69. See also a4sbc 1945 and ra4sbc 1997.

Assertion

Ref	Expression
stdpc4	|- (A.xph -> [y / x]ph)

Proof of Theorem stdpc4

Step	Hyp	Ref	Expression
1		ax-1 4	. . 3 |- (ph -> (x = y -> ph))
2	1	19.20i 992	. 2 |- (A.xph -> A.x(x = y -> ph))
3		sb2 1177	. 2 |- (A.x(x = y -> ph) -> [y / x]ph)
4	2, 3	syl 10	1 |- (A.xph -> [y / x]ph)

Syntax hints:   -> wi 3  A.wal 954  [wsbc 1170

This theorem is referenced by:  sbf 1186  hbs1f 1189  a4sbe 1243  a4sbim 1244  a4sbbi 1245  sb8 1261  sb9i 1263  a4sbc 1945  ra4sbc 1997  nd1 4930  nd2 4931

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

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