Theorem dvelim 2017

Description: This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" -. A. x x = 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. x A. z, conjoin them, and apply dvelimdf 2016.

Other variants of this theorem are dvelimf 2015 (with no distinct variable restrictions) and dvelimALT 2010 (that avoids ax-10 1505). (Contributed by NM, 23-Nov-1994.)

Hypotheses

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

Assertion

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

Distinct variable group:   ps, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y)

Proof of Theorem dvelim

Step	Hyp	Ref	Expression
1		dvelim.1	. 2 |- ( ph -> A. x ph )
2		ax-17 1526	. 2 |- ( ps -> A. z ps )
3		dvelim.2	. 2 |- ( z = y -> ( ph <-> ps ) )
4	1, 2, 3	dvelimf 2015	1 |- ( -. A. x x = y -> ( ps -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by: (None)