Theorem eliminable-abeqv 36237

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 2717	. 2 ⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦))
2		eliminable-velab 36235	. . . 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 1813	. 2 ⊢ (∀𝑧(𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ 𝑦))
5	1, 4	bitri 275	1 ⊢ ({𝑥 ∣ 𝜑} = 𝑦 ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 ∈ 𝑦))

Syntax hints:   ↔ wb 205  ∀wal 1531   = wceq 1533  [wsb 2059   ∈ wcel 2098  {cab 2701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-clab 2702  df-cleq 2716

This theorem is referenced by: (None)