Theorem bnj1436 31427

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 2912	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	2	biimpi 208	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  {cab 2785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-tru 1657  df-ex 1876  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795

This theorem is referenced by:  bnj1517  31437  bnj66  31447  bnj900  31516  bnj1296  31606  bnj1371  31614  bnj1374  31616  bnj1398  31619  bnj1450  31635  bnj1497  31645  bnj1498  31646  bnj1514  31648  bnj1501  31652