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Theorem bnj1465 32016
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1465.1 (𝑥 = 𝐴 → (𝜑𝜓))
bnj1465.2 (𝜓 → ∀𝑥𝜓)
bnj1465.3 (𝜒𝜓)
Assertion
Ref Expression
bnj1465 ((𝜒𝐴𝑉) → ∃𝑥𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem bnj1465
StepHypRef Expression
1 bnj1465.3 . . . 4 (𝜒𝜓)
21adantr 481 . . 3 ((𝜒𝐴𝑉) → 𝜓)
3 bnj1465.2 . . . . 5 (𝜓 → ∀𝑥𝜓)
4 bnj1465.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4bnj1464 32015 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
65adantl 482 . . 3 ((𝜒𝐴𝑉) → ([𝐴 / 𝑥]𝜑𝜓))
72, 6mpbird 258 . 2 ((𝜒𝐴𝑉) → [𝐴 / 𝑥]𝜑)
87spesbcd 3863 1 ((𝜒𝐴𝑉) → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770
This theorem is referenced by:  bnj1463  32224
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