Theorem eliminable-veqab 37220

Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment): variable equals abstraction. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eliminable-veqab

Step	Hyp	Ref	Expression
1		dfcleq 2733	. 2 ⊢ (𝑥 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}))
2		eliminable-velab 37219	. . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
3	2	bibi2i 338	. . 3 ⊢ ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 ∈ 𝑥 ↔ [𝑧 / 𝑦]𝜑))
4	3	albii 1826	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ [𝑧 / 𝑦]𝜑))
5	1, 4	bitri 276	1 ⊢ (𝑥 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ [𝑧 / 𝑦]𝜑))

Syntax hints:   ↔ wb 207  ∀wal 1545   = wceq 1547  [wsb 2073   ∈ wcel 2119  {cab 2718

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-clab 2719  df-cleq 2732

This theorem is referenced by:  eliminable-abelv  37223  eliminable-abelab  37224