Theorem rgenm 3343

Description: Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)

Hypothesis

Ref	Expression
rgenm.1	⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑)

Assertion

Ref	Expression
rgenm	⊢ ∀𝑥 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rgenm

Step	Hyp	Ref	Expression
1		nfe1 1425	. . . . 5 ⊢ Ⅎ𝑥∃𝑥 𝑥 ∈ 𝐴
2		rgenm.1	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑)
3	2	ex 113	. . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑))
4	1, 3	alrimi 1455	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5		19.38 1606	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)))
6	4, 5	ax-mp 7	. . 3 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑))
7		pm5.4 247	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → 𝜑))
8	7	albii 1399	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
9	6, 8	mpbi 143	. 2 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)
10		df-ral 2353	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
11	9, 10	mpbir 144	1 ⊢ ∀𝑥 ∈ 𝐴 𝜑

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1282  ∃wex 1421   ∈ wcel 1433  ∀wral 2348

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353

This theorem is referenced by: (None)