Theorem dfsb7 1379

Description: An alternate definition of proper substitution df-sb 1209. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and ph of the definiendum. No distinct variable conflicts arise because z effectively insulates x from y. To achieve this, we use a chain of two substitutions in the form of sb5 1306, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 1506. Theorem sb7f 1380 provides a version where ph and z don't have to be distinct.

Assertion

Ref	Expression
dfsb7	|- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))

Distinct variable groups:   x,z   y,z   ph,z

Proof of Theorem dfsb7

Step	Hyp	Ref	Expression
1		sb5 1306	. . 3 |- ([z / x]ph <-> E.x(x = z /\ ph))
2	1	sbbii 1211	. 2 |- ([y / z][z / x]ph <-> [y / z]E.x(x = z /\ ph))
3		ax-17 1007	. . 3 |- (ph -> A.zph)
4	3	sbco2 1293	. 2 |- ([y / z][z / x]ph <-> [y / x]ph)
5		sb5 1306	. 2 |- ([y / z]E.x(x = z /\ ph) <-> E.z(z = y /\ E.x(x = z /\ ph)))
6	2, 4, 5	3bitr3i 179	1 |- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))

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

This theorem is referenced by:  sb7f 1380

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209

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