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Theorem sbcexgOLD 37774
 Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcex 3412 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcexgOLD (𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcexgOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3405 . 2 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
2 dfsbcq2 3405 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
32exbidv 1837 . 2 (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
4 sbex 2451 . 2 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
51, 3, 4vtoclbg 3240 1 (𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃wex 1695  [wsb 1867   ∈ wcel 1977  [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-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  csbunigOLD  38073  csbxpgOLD  38075  csbrngOLD  38078  onfrALTlem5VD  38143  csbxpgVD  38152  csbrngVD  38154  csbunigVD  38156
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