Theorem bnj1476 34984

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1476.1	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
bnj1476.2	⊢ (𝜓 → 𝐷 = ∅)

Assertion

Ref	Expression
bnj1476	⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑)

Proof of Theorem bnj1476

Step	Hyp	Ref	Expression
1		bnj1476.2	. . . 4 ⊢ (𝜓 → 𝐷 = ∅)
2		bnj1476.1	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
3		nfrab1 3420	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
4	2, 3	nfcxfr 2897	. . . . 5 ⊢ Ⅎ𝑥𝐷
5	4	eq0f 4300	. . . 4 ⊢ (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐷)
6	1, 5	sylib 218	. . 3 ⊢ (𝜓 → ∀𝑥 ¬ 𝑥 ∈ 𝐷)
7	2	reqabi 3423	. . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
8	7	notbii 320	. . . 4 ⊢ (¬ 𝑥 ∈ 𝐷 ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
9		iman 401	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
10	8, 9	sylbb2 238	. . 3 ⊢ (¬ 𝑥 ∈ 𝐷 → (𝑥 ∈ 𝐴 → 𝜑))
11	6, 10	sylg 1825	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
12	11	bnj1142 34926	1 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rab 3401  df-dif 3905  df-nul 4287

This theorem is referenced by:  bnj1312  35195