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Theorem ralbida 2363
Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
ralbida.1 𝑥𝜑
ralbida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralbida (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Proof of Theorem ralbida
StepHypRef Expression
1 ralbida.1 . . 3 𝑥𝜑
2 ralbida.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32pm5.74da 432 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3albid 1547 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 2354 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
6 df-ral 2354 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 221 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wnf 1390  wcel 1434  wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354
This theorem is referenced by:  ralbidva  2365  ralbid  2367  2ralbida  2388  ralbi  2490  caucvgsrlemgt1  7033
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