Theorem abeq2d 2864

Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2862). (Contributed by NM, 16-Nov-1995.)

Hypothesis

Ref	Expression
abeq2d.1	⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Assertion

Ref	Expression
abeq2d	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Proof of Theorem abeq2d

Step	Hyp	Ref	Expression
1		abeq2d.1	. . 3 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
2	1	eleq2d 2816	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}))
3		abid 2718	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
4	2, 3	bitrdi 290	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1543   ∈ wcel 2112  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809

This theorem is referenced by:  abeq2i  2865  fvelimab  6762  mapsnend  8691  nosupbnd2  33605  noinfbnd2  33620  fvineqsneu  35268  fvineqsneq  35269  ispridlc  35914  ac6s6  36016  dib1dim  38865  prprspr2  44586