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Theorem rexbida 2629
 Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
ralbida.1 xφ
ralbida.2 ((φ x A) → (ψχ))
Assertion
Ref Expression
rexbida (φ → (x A ψx A χ))

Proof of Theorem rexbida
StepHypRef Expression
1 ralbida.1 . . 3 xφ
2 ralbida.2 . . . 4 ((φ x A) → (ψχ))
32pm5.32da 622 . . 3 (φ → ((x A ψ) ↔ (x A χ)))
41, 3exbid 1773 . 2 (φ → (x(x A ψ) ↔ x(x A χ)))
5 df-rex 2620 . 2 (x A ψx(x A ψ))
6 df-rex 2620 . 2 (x A χx(x A χ))
74, 5, 63bitr4g 279 1 (φ → (x A ψx A χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2620 This theorem is referenced by:  rexbidva  2631  rexbid  2633  dfiun2g  3999  fun11iun  5305
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