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Theorem sb8e 2558
Description: Substitution of variable in existential quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2368. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
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 2525 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2246 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvex 2413 1 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wex 1773  wnf 1777  [wsb 2062
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 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063
This theorem is referenced by:  2sb8e  2574  sb8mo  2686  bnj985  32130  sbcexf  35280  exlimddvfi  35287  pm11.58  40606
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