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Theorem spcimedv 2816
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1 (𝜑𝐴𝐵)
spcimedv.2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
Assertion
Ref Expression
spcimedv (𝜑 → (𝜒 → ∃𝑥𝜓))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcimedv
StepHypRef Expression
1 spcimedv.2 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
21ex 114 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜒𝜓)))
32alrimiv 1867 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)))
4 spcimdv.1 . 2 (𝜑𝐴𝐵)
5 nfv 1521 . . 3 𝑥𝜒
6 nfcv 2312 . . 3 𝑥𝐴
75, 6spcimegft 2808 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)) → (𝐴𝐵 → (𝜒 → ∃𝑥𝜓)))
83, 4, 7sylc 62 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1346   = wceq 1348  wex 1485  wcel 2141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732
This theorem is referenced by:  rspcimedv  2836  fihashf1rn  10723
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