Theorem bnj1212 32779

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1212.1	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
bnj1212.2	⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))

Assertion

Ref	Expression
bnj1212	⊢ (𝜃 → 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)   𝜃(𝑥)   𝜏(𝑥)   𝐵(𝑥)

Proof of Theorem bnj1212

Step	Hyp	Ref	Expression
1		bnj1212.1	. . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
2	1	ssrab3 4015	. 2 ⊢ 𝐵 ⊆ 𝐴
3		bnj1212.2	. . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))
4	3	simp2bi 1145	. 2 ⊢ (𝜃 → 𝑥 ∈ 𝐵)
5	2, 4	bnj1213 32778	1 ⊢ (𝜃 → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {crab 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by:  bnj1204  32992  bnj1296  33001  bnj1415  33018  bnj1421  33022  bnj1442  33029  bnj1452  33032  bnj1489  33036