Theorem eqrdav 2739

Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)

Hypotheses

Ref	Expression
eqrdav.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶)
eqrdav.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)
eqrdav.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Assertion

Ref	Expression
eqrdav	⊢ (𝜑 → 𝐴 = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem eqrdav

Step	Hyp	Ref	Expression
1		eqrdav.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶)
2		eqrdav.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)
3		eqrdav.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
4	1, 2, 3	bibiad 845	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	eqrdv 2738	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119

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-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2732

This theorem is referenced by:  boxcutc  8886  supminf  12883  f1omvdconj  19419  fmucndlem  24280  lsmsnorb  33481  ballotlemsima  34707  supminfxr  45908