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Theorem bnj1454 30166
 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 3403 . . 3 ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑})
21a1i 11 . 2 (𝐵 ∈ V → ([𝐵 / 𝑥]𝜑𝐵 ∈ {𝑥𝜑}))
3 bnj1454.1 . . 3 𝐴 = {𝑥𝜑}
43eleq2i 2680 . 2 (𝐵𝐴𝐵 ∈ {𝑥𝜑})
52, 4syl6rbbr 278 1 (𝐵 ∈ V → (𝐵𝐴[𝐵 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-sbc 3403 This theorem is referenced by:  bnj1452  30374  bnj1463  30377
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