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Theorem bnj1454 34810
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1454.1 𝐴 = {𝑥𝜑}
Assertion
Ref Expression
bnj1454 (𝐵 ∈ V → (𝐵𝐴[𝐵 / 𝑥]𝜑))

Proof of Theorem bnj1454
StepHypRef Expression
1 bnj1454.1 . . 3 𝐴 = {𝑥𝜑}
21eleq2i 2830 . 2 (𝐵𝐴𝐵 ∈ {𝑥𝜑})
3 df-sbc 3799 . . 3 ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑})
43a1i 11 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑}))
52, 4bitr4id 290 1 (𝐵 ∈ V → (𝐵𝐴[𝐵 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2103  {cab 2711  Vcvv 3482  [wsbc 3798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2726  df-clel 2813  df-sbc 3799
This theorem is referenced by:  bnj1452  35020  bnj1463  35023
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