Theorem bnj1212 35496

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 4010	. 2 ⊢ 𝐵 ⊆ 𝐴
3		bnj1212.2	. . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏))
4	3	simp2bi 1164	. 2 ⊢ (𝜃 → 𝑥 ∈ 𝐵)
5	2, 4	bnj1213 35495	1 ⊢ (𝜃 → 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {crab 3416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-ss 3901

This theorem is used by:  bnj1204  35709  bnj1296  35718  bnj1415  35735  bnj1421  35739  bnj1442  35746  bnj1452  35749  bnj1489  35753