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Theorem raleqtrdv 3328
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 3326 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
41, 3mpbid 232 1 (𝜑 → ∀𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-ral 3062  df-rex 3071
This theorem is referenced by:  subgpgp  19615  znf1o  21570  perfopn  23193  ordtt1  23387  tx1stc  23658  xkococnlem  23667  dgrlem  26268  dchrisum0flb  27554  wlknewwlksn  29907  sigaclcu3  34123  subfacp1lem3  35187  poimirlem1  37628
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