Theorem bnj1436 34601

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 2869	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	2	biimpi 215	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {cab 2702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802

This theorem is referenced by:  bnj1517  34612  bnj66  34622  bnj900  34691  bnj1296  34783  bnj1371  34791  bnj1374  34793  bnj1398  34796  bnj1450  34812  bnj1497  34822  bnj1498  34823  bnj1514  34825  bnj1501  34829