Theorem bnj1230 35298

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

Hypothesis

Ref	Expression
bnj1230.1	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}

Assertion

Ref	Expression
bnj1230	⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)

Proof of Theorem bnj1230

Step	Hyp	Ref	Expression
1		bnj1230.1	. . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
2		nfrab1 3434	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
3	1, 2	nfcxfr 2922	. 2 ⊢ Ⅎ𝑥𝐵
4	3	nfcrii 2919	1 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {crab 3414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3415

This theorem is used by:  bnj1312  35554