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Theorem rspcimdv 2625
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1 (φA B)
rspcimdv.2 ((φ x = A) → (ψχ))
Assertion
Ref Expression
rspcimdv (φ → (x B ψχ))
Distinct variable groups:   x,A   x,B   φ,x   χ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem rspcimdv
StepHypRef Expression
1 df-ral 2280 . 2 (x B ψx(x Bψ))
2 rspcimdv.1 . . 3 (φA B)
3 simpr 103 . . . . . . 7 ((φ x = A) → x = A)
43eleq1d 2079 . . . . . 6 ((φ x = A) → (x BA B))
54biimprd 147 . . . . 5 ((φ x = A) → (A Bx B))
6 rspcimdv.2 . . . . 5 ((φ x = A) → (ψχ))
75, 6imim12d 68 . . . 4 ((φ x = A) → ((x Bψ) → (A Bχ)))
82, 7spcimdv 2605 . . 3 (φ → (x(x Bψ) → (A Bχ)))
92, 8mpid 37 . 2 (φ → (x(x Bψ) → χ))
101, 9syl5bi 141 1 (φ → (x B ψχ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1221   = wceq 1223   wcel 1366  wral 2275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-v 2528
This theorem is referenced by:  rspcdv  2627
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