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Theorem bnj538OLD 29857
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Obsolete version of bnj538 29856 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 2900 . . 3 (∀𝑥𝐵 𝜑 ↔ ∀𝑥(𝑥𝐵𝜑))
21sbcbii 3457 . 2 ([𝐴 / 𝑦]𝑥𝐵 𝜑[𝐴 / 𝑦]𝑥(𝑥𝐵𝜑))
3 bnj538OLD.1 . . . . . 6 𝐴 ∈ V
4 sbcimg 3443 . . . . . 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 29854 . . . . . 6 ([𝐴 / 𝑦]𝑥𝐵𝑥𝐵)
76imbi1i 337 . . . . 5 (([𝐴 / 𝑦]𝑥𝐵[𝐴 / 𝑦]𝜑) ↔ (𝑥𝐵[𝐴 / 𝑦]𝜑))
85, 7bitri 262 . . . 4 ([𝐴 / 𝑦](𝑥𝐵𝜑) ↔ (𝑥𝐵[𝐴 / 𝑦]𝜑))
98albii 1736 . . 3 (∀𝑥[𝐴 / 𝑦](𝑥𝐵𝜑) ↔ ∀𝑥(𝑥𝐵[𝐴 / 𝑦]𝜑))
10 sbcal 3451 . . 3 ([𝐴 / 𝑦]𝑥(𝑥𝐵𝜑) ↔ ∀𝑥[𝐴 / 𝑦](𝑥𝐵𝜑))
11 df-ral 2900 . . 3 (∀𝑥𝐵 [𝐴 / 𝑦]𝜑 ↔ ∀𝑥(𝑥𝐵[𝐴 / 𝑦]𝜑))
129, 10, 113bitr4i 290 . 2 ([𝐴 / 𝑦]𝑥(𝑥𝐵𝜑) ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑)
132, 12bitri 262 1 ([𝐴 / 𝑦]𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 [𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wcel 1976  wral 2895  Vcvv 3172  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-ral 2900  df-v 3174  df-sbc 3402
This theorem is referenced by: (None)
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