Theorem bnj1454 32822

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

Hypothesis

Ref	Expression
bnj1454.1	⊢ 𝐴 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
bnj1454	⊢ (𝐵 ∈ V → (𝐵 ∈ 𝐴 ↔ [𝐵 / 𝑥]𝜑))

Proof of Theorem bnj1454

Step	Hyp	Ref	Expression
1		bnj1454.1	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
2	1	eleq2i 2830	. 2 ⊢ (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
3		df-sbc 3717	. . 3 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
4	3	a1i 11	. 2 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
5	2, 4	bitr4id 290	1 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝐴 ↔ [𝐵 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {cab 2715  Vcvv 3432  [wsbc 3716

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-ex 1783  df-cleq 2730  df-clel 2816  df-sbc 3717

This theorem is referenced by:  bnj1452  33032  bnj1463  33035