Theorem bnj1436 34832

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

Hypothesis

Ref	Expression
bnj1436.1	⊢ 𝐴 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
bnj1436	⊢ (𝑥 ∈ 𝐴 → 𝜑)

Proof of Theorem bnj1436

Step	Hyp	Ref	Expression
1		bnj1436.1	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
2	1	eqabri 2883	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	2	biimpi 216	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  {cab 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814

This theorem is referenced by:  bnj1517  34843  bnj66  34853  bnj900  34922  bnj1296  35014  bnj1371  35022  bnj1374  35024  bnj1398  35027  bnj1450  35043  bnj1497  35053  bnj1498  35054  bnj1514  35056  bnj1501  35060