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Theorem rexeqtrrdv 3331
Description: Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
Hypotheses
Ref Expression
rexeqtrrdv.1 (𝜑 → ∃𝑥𝐴 𝜓)
rexeqtrrdv.2 (𝜑𝐵 = 𝐴)
Assertion
Ref Expression
rexeqtrrdv (𝜑 → ∃𝑥𝐵 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rexeqtrrdv
StepHypRef Expression
1 rexeqtrrdv.1 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
2 rexeqtrrdv.2 . . 3 (𝜑𝐵 = 𝐴)
32rexeqdv 3327 . 2 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑥𝐴 𝜓))
41, 3mpbird 257 1 (𝜑 → ∃𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wrex 3070
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-rex 3071
This theorem is referenced by:  zornn0g  10545  ablfacrplem  20085  ablfac2  20109  2ndcctbss  23463  1stcelcls  23469  wwlksnextsurj  29920  primrootsunit1  42098
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