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Theorem exintrbi 1597
Description: Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
Assertion
Ref Expression
exintrbi (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))

Proof of Theorem exintrbi
StepHypRef Expression
1 pm4.71 386 . . 3 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
21albii 1431 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ (𝜑𝜓)))
3 exbi 1568 . 2 (∀𝑥(𝜑 ↔ (𝜑𝜓)) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))
42, 3sylbi 120 1 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1314  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  exintr  1598
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