Theorem dvelim 1350

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 1249.

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-17 969	. 2 |- (ps -> A.zps)
3		dvelim.2	. 2 |- (z = y -> (ph <-> ps))
4	1, 2, 3	dvelimf 1248	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 952   = wceq 954

This theorem is referenced by:  rgen2a 1696  ralcom2 1773

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170

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