Theorem sb6 1305

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 1304	. . 3 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
2	1	anbi2i 483	. 2 |- (((x = y -> ph) /\ E.x(x = y /\ ph)) <-> ((x = y -> ph) /\ A.x(x = y -> ph)))
3		df-sb 1209	. 2 |- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
4		ax-4 1009	. . 3 |- (A.x(x = y -> ph) -> (x = y -> ph))
5	4	pm4.71ri 641	. 2 |- (A.x(x = y -> ph) <-> ((x = y -> ph) /\ A.x(x = y -> ph)))
6	2, 3, 5	3bitr4i 181	1 |- ([y / x]ph <-> A.x(x = y -> ph))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016  [wsbc 1207

This theorem is referenced by:  sb5 1306  2sb6 1375  sb6a 1376  exsb 1389  sbal2 1397

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  ax-16 1247  ax-11o 1255

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

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