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Theorem rspesbca 3813
Description: Existence form of rspsbca 3812. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
rspesbca ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rspesbca
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3726 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
21rspcev 3574 . 2 ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦𝐵 [𝑦 / 𝑥]𝜑)
3 cbvrexsvw 3418 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑦 / 𝑥]𝜑)
42, 3sylibr 237 1 ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  [wsb 2069  wcel 2112  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-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-sbc 3724
This theorem is referenced by:  spesbc  3814  2nreu  4352  rexopabb  5383  indexfi  8820  indexdom  35165
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