Theorem ralf0 2363

Description: The quantification of a falsehood is vacuous when true.

Hypothesis

Ref	Expression
ralf0.1	⊢ ¬ φ

Assertion

Ref	Expression
ralf0	⊢ (∀x ∈ A φ ↔ A = ∅)

Distinct variable group:   x,A

Proof of Theorem ralf0

Step	Hyp	Ref	Expression
1		ralf0.1	. . . . 5 ⊢ ¬ φ
2		con3 94	. . . . 5 ⊢ ((x ∈ A → φ) → (¬ φ → ¬ x ∈ A))
3	1, 2	mpi 44	. . . 4 ⊢ ((x ∈ A → φ) → ¬ x ∈ A)
4	3	19.20i 994	. . 3 ⊢ (∀x(x ∈ A → φ) → ∀x ¬ x ∈ A)
5		df-ral 1652	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6		eq0 2298	. . 3 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A)
7	4, 5, 6	3imtr4 219	. 2 ⊢ (∀x ∈ A φ → A = ∅)
8		rzal 2359	. 2 ⊢ (A = ∅ → ∀x ∈ A φ)
9	7, 8	impbi 157	1 ⊢ (∀x ∈ A φ ↔ A = ∅)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146  ∀wal 956   = wceq 958   ∈ wcel 960  ∀wral 1648  ∅c0 2283

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-nul 2284

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