Theorem spcv 2919

Description: Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)

Hypotheses

Ref	Expression
spcv.1	|- A e. _V
spcv.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
spcv	|- ( A. x ph -> ps )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem spcv

Step	Hyp	Ref	Expression
1		spcv.1	. 2 |- A e. _V
2		spcv.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	spcgv 2912	. 2 |- ( A e. _V -> ( A. x ph -> ps ) )
4	1, 3	ax-mp 5	1 |- ( A. x ph -> ps )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   e. wcel 2209   _Vcvv 2821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is used by:  morex  3010  exmidexmid  4335  exmidsssn  4341  exmidel  4344  rext  4357  ontr2exmid  4674  regexmidlem1  4682  reg2exmid  4685  relop  4932  uchoice  6375  disjxp1  6476  rdgtfr  6649  ssfiexmid  7182  ssfiexmidt  7184  domfiexmid  7186  diffitest  7195  findcard  7196  exmidpw2en  7223  fiintim  7242  fisseneq  7246  finomni  7486  exmidomni  7488  exmidlpo  7489  ballotfilem2  13286  exmidunben  13375  ivthreinc  15871  bj-d0clsepcl  17156  bj-inf2vnlem1  17201  subctctexmid  17235  wexmiddiffilem  17248  wexmiddifxy  17251