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Theorem reubidv 2510
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
Hypothesis
Ref Expression
reubidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
reubidv (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reubidv
StepHypRef Expression
1 reubidv.1 . . 3 (𝜑 → (𝜓𝜒))
21adantr 265 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32reubidva 2509 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wcel 1409  ∃!wreu 2325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-eu 1919  df-reu 2330
This theorem is referenced by:  reueqd  2532  sbcreug  2866  srpospr  6925  creur  7987  creui  7988  divalg2  10238
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