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Theorem bnj1465 30169
 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 480 . . 3 ((𝜒𝐴𝑉) → 𝜓)
3 bnj1465.2 . . . . 5 (𝜓 → ∀𝑥𝜓)
4 bnj1465.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4bnj1464 30168 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
65adantl 481 . . 3 ((𝜒𝐴𝑉) → ([𝐴 / 𝑥]𝜑𝜓))
72, 6mpbird 246 . 2 ((𝜒𝐴𝑉) → [𝐴 / 𝑥]𝜑)
87spesbcd 3488 1 ((𝜒𝐴𝑉) → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403 This theorem is referenced by:  bnj1463  30377
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