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Theorem bnj1454 32013
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 df-sbc 3770 . . 3 ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑})
21a1i 11 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑}))
3 bnj1454.1 . . 3 𝐴 = {𝑥𝜑}
43eleq2i 2901 . 2 (𝐵𝐴𝐵 ∈ {𝑥𝜑})
52, 4syl6rbbr 291 1 (𝐵 ∈ V → (𝐵𝐴[𝐵 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {cab 2796  Vcvv 3492  [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-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890  df-sbc 3770
This theorem is referenced by:  bnj1452  32221  bnj1463  32224
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