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Theorem ralbidv2 2414
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)
Hypothesis
Ref Expression
ralbidv2.1 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
ralbidv2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ralbidv2
StepHypRef Expression
1 ralbidv2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
21albidv 1778 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑥(𝑥𝐵𝜒)))
3 df-ral 2396 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
4 df-ral 2396 . 2 (∀𝑥𝐵 𝜒 ↔ ∀𝑥(𝑥𝐵𝜒))
52, 3, 43bitr4g 222 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312  wcel 1463  wral 2391
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 1406  ax-gen 1408  ax-17 1489
This theorem depends on definitions:  df-bi 116  df-ral 2396
This theorem is referenced by:  ralss  3131  dfsmo2  6150  raluz  9322  isprm3  11695  metcnp  12576
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