Theorem sb6 1262

Description: Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70.

Assertion

Ref	Expression
sb6	|- ([y / x]ph <-> A.x(x = y -> ph))

Distinct variable group:   x,y

Proof of Theorem sb6

Step	Hyp	Ref	Expression
1		sb56 1261	. . 3 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
2	1	anbi2i 479	. 2 |- (((x = y -> ph) /\ E.x(x = y /\ ph)) <-> ((x = y -> ph) /\ A.x(x = y -> ph)))
3		df-sb 1168	. 2 |- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
4		ax-4 970	. . 3 |- (A.x(x = y -> ph) -> (x = y -> ph))
5	4	pm4.71ri 636	. 2 |- (A.x(x = y -> ph) <-> ((x = y -> ph) /\ A.x(x = y -> ph)))
6	2, 3, 5	3bitr4 183	1 |- ([y / x]ph <-> A.x(x = y -> ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  [wsbc 1166

This theorem is referenced by:  sb5 1263  2sb6 1331  sb6a 1332  exsb 1345  sbal2 1351

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168

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