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Theorem raleqtrdv 3326
Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
Hypotheses
Ref Expression
raleqtrdv.1 (𝜑 → ∀𝑥𝐴 𝜓)
raleqtrdv.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
raleqtrdv (𝜑 → ∀𝑥𝐵 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem raleqtrdv
StepHypRef Expression
1 raleqtrdv.1 . 2 (𝜑 → ∀𝑥𝐴 𝜓)
2 raleqtrdv.2 . . 3 (𝜑𝐴 = 𝐵)
32raleqdv 3324 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
41, 3mpbid 232 1 (𝜑 → ∀𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ral 3060  df-rex 3069
This theorem is referenced by:  subgpgp  19630  znf1o  21588  perfopn  23209  ordtt1  23403  tx1stc  23674  xkococnlem  23683  dgrlem  26283  dchrisum0flb  27569  wlknewwlksn  29917  sigaclcu3  34103  subfacp1lem3  35167  poimirlem1  37608
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