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Theorem bnj1465 31041
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 31040 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
65adantl 481 . . 3 ((𝜒𝐴𝑉) → ([𝐴 / 𝑥]𝜑𝜓))
72, 6mpbird 247 . 2 ((𝜒𝐴𝑉) → [𝐴 / 𝑥]𝜑)
87spesbcd 3555 1 ((𝜒𝐴𝑉) → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469
This theorem is referenced by:  bnj1463  31249
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