Theorem sb6 1267

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 1266	. . 3 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
2	1	anbi2i 480	. 2 |- (((x = y -> ph) /\ E.x(x = y /\ ph)) <-> ((x = y -> ph) /\ A.x(x = y -> ph)))
3		df-sb 1172	. 2 |- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
4		ax-4 973	. . 3 |- (A.x(x = y -> ph) -> (x = y -> ph))
5	4	pm4.71ri 638	. 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 954   = wceq 956  E.wex 980  [wsbc 1170

This theorem is referenced by:  sb5 1268  2sb6 1336  sb6a 1337  exsb 1350  sbal2 1358

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  ax-16 1210  ax-11o 1218

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

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