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Theorem bnj538OLD 30936
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Obsolete version of bnj538 30935 as of 30-Mar-2020. (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj538OLD.1 𝐴 ∈ V
Assertion
Ref Expression
bnj538OLD ([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 bnj538OLD
StepHypRef Expression
1 df-ral 2946 . . 3 (∀𝑥𝐵 𝜑 ↔ ∀𝑥(𝑥𝐵𝜑))
21sbcbii 3524 . 2 ([𝐴 / 𝑦]𝑥𝐵 𝜑[𝐴 / 𝑦]𝑥(𝑥𝐵𝜑))
3 bnj538OLD.1 . . . . . 6 𝐴 ∈ V
4 sbcimg 3510 . . . . . 6 (𝐴 ∈ V → ([𝐴 / 𝑦](𝑥𝐵𝜑) ↔ ([𝐴 / 𝑦]𝑥𝐵[𝐴 / 𝑦]𝜑)))
53, 4ax-mp 5 . . . . 5 ([𝐴 / 𝑦](𝑥𝐵𝜑) ↔ ([𝐴 / 𝑦]𝑥𝐵[𝐴 / 𝑦]𝜑))
63bnj525 30933 . . . . . 6 ([𝐴 / 𝑦]𝑥𝐵𝑥𝐵)
76imbi1i 338 . . . . 5 (([𝐴 / 𝑦]𝑥𝐵[𝐴 / 𝑦]𝜑) ↔ (𝑥𝐵[𝐴 / 𝑦]𝜑))
85, 7bitri 264 . . . 4 ([𝐴 / 𝑦](𝑥𝐵𝜑) ↔ (𝑥𝐵[𝐴 / 𝑦]𝜑))
98albii 1787 . . 3 (∀𝑥[𝐴 / 𝑦](𝑥𝐵𝜑) ↔ ∀𝑥(𝑥𝐵[𝐴 / 𝑦]𝜑))
10 sbcal 3518 . . 3 ([𝐴 / 𝑦]𝑥(𝑥𝐵𝜑) ↔ ∀𝑥[𝐴 / 𝑦](𝑥𝐵𝜑))
11 df-ral 2946 . . 3 (∀𝑥𝐵 [𝐴 / 𝑦]𝜑 ↔ ∀𝑥(𝑥𝐵[𝐴 / 𝑦]𝜑))
129, 10, 113bitr4i 292 . 2 ([𝐴 / 𝑦]𝑥(𝑥𝐵𝜑) ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑)
132, 12bitri 264 1 ([𝐴 / 𝑦]𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946  df-v 3233  df-sbc 3469
This theorem is referenced by: (None)
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