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Theorem cbvexsv 38241
Description: A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cbvexsv (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvexsv
StepHypRef Expression
1 cbvrexsv 3171 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V [𝑦 / 𝑥]𝜑)
2 rexv 3206 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
3 rexv 3206 . 2 (∃𝑦 ∈ V [𝑦 / 𝑥]𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
41, 2, 33bitr3i 290 1 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1701  [wsb 1877  wrex 2908  Vcvv 3186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188
This theorem is referenced by:  onfrALTlem1  38242  onfrALTlem1VD  38606
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