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Theorem sb8eh 1784
 Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb8eh.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
sb8eh (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8eh
StepHypRef Expression
1 sb8eh.1 . 2 (𝜑 → ∀𝑦𝜑)
21hbsb3 1737 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3 sbequ12 1702 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvexh 1686 1 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1288  ∃wex 1427  [wsb 1693 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473 This theorem depends on definitions:  df-bi 116  df-sb 1694 This theorem is referenced by:  exsb  1933
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