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Theorem sb8e 2543
Description: Substitution of variable in existential quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2367. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb5rf.1 𝑦𝜑
Assertion
Ref Expression
sb8e (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8e
StepHypRef Expression
1 sb5rf.1 . 2 𝑦𝜑
21nfs1 2482 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2278 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvex 2411 1 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wex 1875  wnf 1879  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880  df-sb 2065
This theorem is referenced by:  sbnf2OLD  2560  2sb8e  2587  sb8mo  2653  mo3OLD  2659  bnj985  31540  bj-mo3OLD  33327  sbcexf  34405  exlimddvfi  34413  pm11.58  39372
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