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Theorem bnj538 31114
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 3652 . 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 196  wcel 2137  wral 3048  Vcvv 3338  [wsbc 3574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-v 3340  df-sbc 3575
This theorem is referenced by:  bnj92  31237  bnj539  31266  bnj540  31267
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