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Theorem rgenm 3433
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 1455 . . . . 5 𝑥𝑥 𝑥𝐴
2 rgenm.1 . . . . . 6 ((∃𝑥 𝑥𝐴𝑥𝐴) → 𝜑)
32ex 114 . . . . 5 (∃𝑥 𝑥𝐴 → (𝑥𝐴𝜑))
41, 3alrimi 1485 . . . 4 (∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑))
5 19.38 1637 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)))
64, 5ax-mp 5 . . 3 𝑥(𝑥𝐴 → (𝑥𝐴𝜑))
7 pm5.4 248 . . . 4 ((𝑥𝐴 → (𝑥𝐴𝜑)) ↔ (𝑥𝐴𝜑))
87albii 1429 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐴𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
96, 8mpbi 144 . 2 𝑥(𝑥𝐴𝜑)
10 df-ral 2396 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
119, 10mpbir 145 1 𝑥𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312  wex 1451  wcel 1463  wral 2391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-ral 2396
This theorem is referenced by: (None)
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