Theorem bnj1436 32847

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	abeq2i 2870	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	2	biimpi 215	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2101  {cab 2710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-12 2166  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1540  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811

This theorem is referenced by:  bnj1517  32858  bnj66  32868  bnj900  32937  bnj1296  33029  bnj1371  33037  bnj1374  33039  bnj1398  33042  bnj1450  33058  bnj1497  33068  bnj1498  33069  bnj1514  33071  bnj1501  33075