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Theorem bnj538 32018
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020.)
Hypothesis
Ref Expression
bnj538.1 𝐴 ∈ V
Assertion
Ref Expression
bnj538 ([𝐴 / 𝑦]𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑥)

Proof of Theorem bnj538
StepHypRef Expression
1 bnj538.1 . 2 𝐴 ∈ V
2 sbcralg 3832 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑))
31, 2ax-mp 5 1 ([𝐴 / 𝑦]𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2115  wral 3126  Vcvv 3471  [wsbc 3749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-v 3473  df-sbc 3750
This theorem is referenced by:  bnj92  32141  bnj539  32170  bnj540  32171
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