Theorem dvelim 1347

Description: This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" -. A.xx = y as an antecedent. ph normally has z free and can be read ph(z), and ps substitutes y for z and can be read ph(y). We don't require that x and y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with A.xA.z, conjoin them, and apply dvelimdf 1246.

Hypotheses

Ref	Expression
dvelim.1	|- (ph -> A.xph)
dvelim.2	|- (z = y -> (ph <-> ps))

Assertion

Ref	Expression
dvelim	|- (-. A.x x = y -> (ps -> A.xps))

Distinct variable group:   ps,z

Proof of Theorem dvelim

Step	Hyp	Ref	Expression
1		dvelim.1	. 2 |- (ph -> A.xph)
2		ax-17 968	. 2 |- (ps -> A.zps)
3		dvelim.2	. 2 |- (z = y -> (ph <-> ps))
4	1, 2, 3	dvelimf 1245	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 951   = wceq 953

This theorem is referenced by:  rgen2a 1691  ralcom2 1768

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-11o 1213

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

Copyright terms: Public domain