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Theorem rgenm 3323
 Description: Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
Hypothesis
Ref Expression
rgenm.1 ((∃𝑥 𝑥𝐴𝑥𝐴) → 𝜑)
Assertion
Ref Expression
rgenm 𝑥𝐴 𝜑
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rgenm
StepHypRef Expression
1 nfe1 1385 . . . . 5 𝑥𝑥 𝑥𝐴
2 rgenm.1 . . . . . 6 ((∃𝑥 𝑥𝐴𝑥𝐴) → 𝜑)
32ex 108 . . . . 5 (∃𝑥 𝑥𝐴 → (𝑥𝐴𝜑))
41, 3alrimi 1415 . . . 4 (∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑))
5 19.38 1566 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
64, 5ax-mp 7 . . 3 𝑥(𝑥𝐴 → (𝑥𝐴𝜑))
7 pm5.4 238 . . . 4 ((𝑥𝐴 → (𝑥𝐴𝜑)) ↔ (𝑥𝐴𝜑))
87albii 1359 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
96, 8mpbi 133 . 2 𝑥(𝑥𝐴𝜑)
10 df-ral 2311 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
119, 10mpbir 134 1 𝑥𝐴 𝜑
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311 This theorem is referenced by: (None)
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