Theorem stdpc4 1222

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 1990 and ra4sbc 2047.

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 1028	. 2 |- (A.xph -> A.x(x = y -> ph))
3		sb2 1214	. 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 990  [wsbc 1207

This theorem is referenced by:  sbf 1223  hbs1f 1226  a4sbe 1280  a4sbim 1281  a4sbbi 1282  sbf3t 1285  sb8 1299  sb9i 1301  a4sbc 1990  ra4sbc 2047  nd1 5092  nd2 5093

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 999  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209

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