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Theorem rspesbca 3737
Description: Existence form of rspsbca 3736. (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 3655 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
21rspcev 3511 . 2 ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦𝐵 [𝑦 / 𝑥]𝜑)
3 cbvrexsv 3379 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑦 / 𝑥]𝜑)
42, 3sylibr 226 1 ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  [wsb 2011  wcel 2107  wrex 3091  [wsbc 3652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-sbc 3653
This theorem is referenced by:  spesbc  3738  indexfi  8564  indexdom  34159
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