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Theorem sb8e 2521
Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2375. For a version requiring disjoint variables, but fewer axioms, see sb8ef 2356. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.)
Hypothesis
Ref Expression
sb8.1 𝑦𝜑
Assertion
Ref Expression
sb8e (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8e
StepHypRef Expression
1 sb8.1 . 2 𝑦𝜑
21nfs1 2491 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2249 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvex 2402 1 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wex 1776  wnf 1780  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by:  2sb8e  2533  sb8mo  2599  bnj985  34947  exlimddvfi  38109  pm11.58  44386
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