Theorem bnj1212 34792

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 4092	. 2 ⊢ 𝐵 ⊆ 𝐴
3		bnj1212.2	. . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))
4	3	simp2bi 1145	. 2 ⊢ (𝜃 → 𝑥 ∈ 𝐵)
5	2, 4	bnj1213 34791	1 ⊢ (𝜃 → 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {crab 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-ss 3980

This theorem is referenced by:  bnj1204  35005  bnj1296  35014  bnj1415  35031  bnj1421  35035  bnj1442  35042  bnj1452  35045  bnj1489  35049