Theorem bnj1436 35236

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 2904	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	2	biimpi 219	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  {cab 2740

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837

This theorem is used by:  bnj1517  35247  bnj66  35257  bnj900  35326  bnj1296  35418  bnj1371  35426  bnj1374  35428  bnj1398  35431  bnj1450  35447  bnj1497  35457  bnj1498  35458  bnj1514  35460  bnj1501  35464