Theorem ralf0OLD 4079

Description: Obsolete proof of ralf0 4078 as of 14-Jul-2021. (Contributed by NM, 26-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
ralf0.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
ralf0OLD	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralf0OLD

Step	Hyp	Ref	Expression
1		ralf0.1	. . . . 5 ⊢ ¬ 𝜑
2		con3 149	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴))
3	1, 2	mpi 20	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ¬ 𝑥 ∈ 𝐴)
4	3	alimi 1739	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
5		df-ral 2917	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		eq0 3929	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
7	4, 5, 6	3imtr4i 281	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = ∅)
8		rzal 4073	. 2 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
9	7, 8	impbii 199	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)