Theorem eqri 3935

Description: Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)

Hypotheses

Ref	Expression
eqri.1	⊢ Ⅎ𝑥𝐴
eqri.2	⊢ Ⅎ𝑥𝐵
eqri.3	⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
eqri	⊢ 𝐴 = 𝐵

Proof of Theorem eqri

Step	Hyp	Ref	Expression
1		nftru 1811	. . 3 ⊢ Ⅎ𝑥⊤
2		eqri.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		eqri.2	. . 3 ⊢ Ⅎ𝑥𝐵
4		eqri.3	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
5	4	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
6	1, 2, 3, 5	eqrd 3934	. 2 ⊢ (⊤ → 𝐴 = 𝐵)
7	6	mptru 1554	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 207   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-cleq 2731  df-clel 2814  df-nfc 2888

This theorem is referenced by:  iunab  4981  iinab  4997  rnep  5869  difrab2  32585  esum2dlem  34276  eulerpartlemn  34565