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Theorem cbvexsv 44567
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 3367 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V [𝑦 / 𝑥]𝜑)
2 rexv 3509 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
3 rexv 3509 . 2 (∃𝑦 ∈ V [𝑦 / 𝑥]𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
41, 2, 33bitr3i 301 1 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wex 1779  [wsb 2064  wrex 3070  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482
This theorem is referenced by:  onfrALTlem1  44568  onfrALTlem1VD  44910
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