Theorem bnj1436 32113

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 2950	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	2	biimpi 218	1 ⊢ (𝑥 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  {cab 2801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895

This theorem is referenced by:  bnj1517  32124  bnj66  32134  bnj900  32203  bnj1296  32295  bnj1371  32303  bnj1374  32305  bnj1398  32308  bnj1450  32324  bnj1497  32334  bnj1498  32335  bnj1514  32337  bnj1501  32341