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Theorem ralbida 2432
 Description: Formula-building rule for restricted universal quantifier (deduction form). (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 440 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3albid 1595 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 2422 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
6 df-ral 2422 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
74, 5, 63bitr4g 222 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  Ⅎwnf 1437   ∈ wcel 1481  ∀wral 2417 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 1424  ax-gen 1426  ax-4 1488 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2422 This theorem is referenced by:  ralbidva  2434  ralbid  2436  2ralbida  2457  ralbi  2565  ismkvnex  7033  caucvgsrlemgt1  7623  iswomninnlem  13400
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