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Theorem bnj1454 31250
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 3588 . . 3 ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑})
21a1i 11 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑}))
3 bnj1454.1 . . 3 𝐴 = {𝑥𝜑}
43eleq2i 2842 . 2 (𝐵𝐴𝐵 ∈ {𝑥𝜑})
52, 4syl6rbbr 279 1 (𝐵 ∈ V → (𝐵𝐴[𝐵 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {cab 2757  Vcvv 3351  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764  df-clel 2767  df-sbc 3588
This theorem is referenced by:  bnj1452  31458  bnj1463  31461
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