Theorem ralf0 2355

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 990	. . 3 ⊢ (∀x(x ∈ A → φ) → ∀x ¬ x ∈ A)
5		df-ral 1646	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6		eq0 2290	. . 3 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A)
7	4, 5, 6	3imtr4 219	. 2 ⊢ (∀x ∈ A φ → A = ∅)
8		rzal 2351	. 2 ⊢ (A = ∅ → ∀x ∈ A φ)
9	7, 8	impbi 157	1 ⊢ (∀x ∈ A φ ↔ A = ∅)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146  ∀wal 952   = wceq 954   ∈ wcel 956  ∀wral 1642  ∅c0 2276

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-nul 2277

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