Theorem dvelim 2045

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 2044.

Other variants of this theorem are dvelimf 2043 (with no distinct variable restrictions) and dvelimALT 2038 (that avoids ax-10 1528). (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 1549	. 2 |- ( ps -> A. z ps )
3		dvelim.2	. 2 |- ( z = y -> ( ph <-> ps ) )
4	1, 2, 3	dvelimf 2043	1 |- ( -. A. x x = y -> ( ps -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1371

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 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by: (None)