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Theorem rspesbca 3876
Description: Existence form of rspsbca 3875. (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 3781 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
21rspcev 3611 . 2 ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦𝐵 [𝑦 / 𝑥]𝜑)
3 cbvrexsvw 3313 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑦 / 𝑥]𝜑)
42, 3sylibr 233 1 ((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  [wsb 2059  wcel 2098  wrex 3067  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-sbc 3779
This theorem is referenced by:  spesbc  3877  2nreu  4445  rexopabb  5534  indexfi  9392  indexdom  37240
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