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Theorem sbcrexgOLD 36163
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3480 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcrexgOLD (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcrexgOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3404 . 2 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑))
2 dfsbcq2 3404 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32rexbidv 3033 . 2 (𝑧 = 𝐴 → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
4 nfcv 2750 . . . 4 𝑥𝐵
5 nfs1v 2424 . . . 4 𝑥[𝑧 / 𝑥]𝜑
64, 5nfrex 2989 . . 3 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 2096 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87rexbidv 3033 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 2395 . 2 ([𝑧 / 𝑥]∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
101, 3, 9vtoclbg 3239 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  [wsb 1866  wcel 1976  wrex 2896  [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-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-sbc 3402
This theorem is referenced by:  2sbcrexOLD  36164  sbc2rexgOLD  36166
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