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Theorem bnj1465 34385
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 34384 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
65adantl 481 . . 3 ((𝜒𝐴𝑉) → ([𝐴 / 𝑥]𝜑𝜓))
72, 6mpbird 257 . 2 ((𝜒𝐴𝑉) → [𝐴 / 𝑥]𝜑)
87spesbcd 3872 1 ((𝜒𝐴𝑉) → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-sbc 3773
This theorem is referenced by:  bnj1463  34595
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