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Theorem exintrbi 1815
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 abai 835 . . 3 ((𝜑𝜓) ↔ (𝜑 ∧ (𝜑𝜓)))
21rbaibr 945 . 2 ((𝜑𝜓) → (𝜑 ↔ (𝜑𝜓)))
32alexbii 1757 1 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  exintr  1816
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