Theorem eliminable-abeqv 35030

Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction equals variable. (Contributed by BJ, 30-Apr-2024.) Beware not to use symmetry of class equality. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
eliminable-abeqv	⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ 𝑦))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eliminable-abeqv

Step	Hyp	Ref	Expression
1		dfcleq 2732	. 2 ⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦))
2		eliminable-velab 35028	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
3	2	bibi1i 338	. . 3 ⊢ ((𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦) ↔ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ 𝑦))
4	3	albii 1825	. 2 ⊢ (∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ 𝑦))
5	1, 4	bitri 274	1 ⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ 𝑦))

Syntax hints:   ↔ wb 205  ∀wal 1539   = wceq 1541  [wsb 2070   ∈ wcel 2109  {cab 2716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-clab 2717  df-cleq 2731

This theorem is referenced by: (None)