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Theorem sbcrex 3807
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcrex ([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)

Proof of Theorem sbcrex
StepHypRef Expression
1 nfcv 2958 . 2 𝑦𝐴
2 sbcrext 3805 . 2 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
31, 2ax-mp 5 1 ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wnfc 2939  wrex 3110  [wsbc 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724
This theorem is referenced by:  2nreu  4352  ac6sfi  8750  csbwrdg  13891  rexfiuz  14703  2sbcrex  39718  sbc2rex  39721  iccelpart  43943
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