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Theorem rgenm 3380
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 1430 . . . . 5 𝑥𝑥 𝑥𝐴
2 rgenm.1 . . . . . 6 ((∃𝑥 𝑥𝐴𝑥𝐴) → 𝜑)
32ex 113 . . . . 5 (∃𝑥 𝑥𝐴 → (𝑥𝐴𝜑))
41, 3alrimi 1460 . . . 4 (∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑))
5 19.38 1611 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
64, 5ax-mp 7 . . 3 𝑥(𝑥𝐴 → (𝑥𝐴𝜑))
7 pm5.4 247 . . . 4 ((𝑥𝐴 → (𝑥𝐴𝜑)) ↔ (𝑥𝐴𝜑))
87albii 1404 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
96, 8mpbi 143 . 2 𝑥(𝑥𝐴𝜑)
10 df-ral 2364 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
119, 10mpbir 144 1 𝑥𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287  wex 1426  wcel 1438  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364
This theorem is referenced by: (None)
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