Theorem bnj1454 30912

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		df-sbc 3436	. . 3 ⊢ ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
2	1	a1i 11	. 2 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝜑 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑}))
3		bnj1454.1	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
4	3	eleq2i 2693	. 2 ⊢ (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ {𝑥 ∣ 𝜑})
5	2, 4	syl6rbbr 279	1 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝐴 ↔ [𝐵 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  bnj1452  31120  bnj1463  31123