Theorem eliminable-abelv 37223

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

Assertion

Ref	Expression
eliminable-abelv	⊢ ({𝑥 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eliminable-abelv

Step	Hyp	Ref	Expression
1		dfclel 2816	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ 𝑦))
2		eliminable-veqab 37220	. . . 4 ⊢ (𝑧 = {𝑥 ∣ 𝜑} ↔ ∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑))
3	2	anbi1i 630	. . 3 ⊢ ((𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ 𝑦) ↔ (∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦))
4	3	exbii 1855	. 2 ⊢ (∃𝑧(𝑧 = {𝑥 ∣ 𝜑} ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑧(∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦))
5	1, 4	bitri 276	1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑦 ↔ ∃𝑧(∀𝑡(𝑡 ∈ 𝑧 ↔ [𝑡 / 𝑥]𝜑) ∧ 𝑧 ∈ 𝑦))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786  [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-8 2121  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  df-clel 2815

This theorem is referenced by: (None)