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Theorem bnj1454 31429
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 3634 . . 3 ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑})
21a1i 11 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑}))
3 bnj1454.1 . . 3 𝐴 = {𝑥𝜑}
43eleq2i 2870 . 2 (𝐵𝐴𝐵 ∈ {𝑥𝜑})
52, 4syl6rbbr 282 1 (𝐵 ∈ V → (𝐵𝐴[𝐵 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  {cab 2785  Vcvv 3385  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2792  df-clel 2795  df-sbc 3634
This theorem is referenced by:  bnj1452  31637  bnj1463  31640
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