Theorem eqrd 3586

Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Hypotheses

Ref	Expression
eqrd.0	⊢ Ⅎ𝑥𝜑
eqrd.1	⊢ Ⅎ𝑥𝐴
eqrd.2	⊢ Ⅎ𝑥𝐵
eqrd.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Assertion

Ref	Expression
eqrd	⊢ (𝜑 → 𝐴 = 𝐵)

Proof of Theorem eqrd

Step	Hyp	Ref	Expression
1		eqrd.0	. . 3 ⊢ Ⅎ𝑥𝜑
2		eqrd.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		eqrd.2	. . 3 ⊢ Ⅎ𝑥𝐵
4		eqrd.3	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	biimpd 218	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6	1, 2, 3, 5	ssrd 3573	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	4	biimprd 237	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
8	1, 3, 2, 7	ssrd 3573	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
9	6, 8	eqssd 3585	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554

This theorem is referenced by:  sniota  5795  dissnlocfin  21142  imasnopn  21303  imasncld  21304  imasncls  21305  blval2  22177  eqri  28735  fimarab  28825  ofpreima  28848  ordtconlem1  29298  qqhval2  29354  bj-sspwpwab  32239  bj-xnex  32245  topdifinfindis  32370  icorempt2  32375  isbasisrelowllem1  32379  isbasisrelowllem2  32380  areaquad  36821  rfcnpre1  38201  rfcnpre2  38213