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Theorem rspcimedv 2626
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1 (φA B)
rspcimedv.2 ((φ x = A) → (χψ))
Assertion
Ref Expression
rspcimedv (φ → (χx B ψ))
Distinct variable groups:   x,A   x,B   φ,x   χ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem rspcimedv
StepHypRef Expression
1 rspcimdv.1 . . 3 (φA B)
2 simpr 103 . . . . . . 7 ((φ x = A) → x = A)
32eleq1d 2079 . . . . . 6 ((φ x = A) → (x BA B))
43biimprd 147 . . . . 5 ((φ x = A) → (A Bx B))
5 rspcimedv.2 . . . . 5 ((φ x = A) → (χψ))
64, 5anim12d 318 . . . 4 ((φ x = A) → ((A B χ) → (x B ψ)))
71, 6spcimedv 2607 . . 3 (φ → ((A B χ) → x(x B ψ)))
81, 7mpand 405 . 2 (φ → (χx(x B ψ)))
9 df-rex 2281 . 2 (x B ψx(x B ψ))
108, 9syl6ibr 151 1 (φ → (χx B ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223  wex 1354   wcel 1366  wrex 2276
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-rex 2281  df-v 2528
This theorem is referenced by:  rspcedv  2628
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