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Theorem eliminable-veqab 34586
 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 = {𝑦𝜑} ↔ ∀𝑧(𝑧𝑥 ↔ [𝑧 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eliminable-veqab
StepHypRef Expression
1 dfcleq 2752 . 2 (𝑥 = {𝑦𝜑} ↔ ∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝜑}))
2 eliminable-velab 34585 . . . 4 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
32bibi2i 342 . . 3 ((𝑧𝑥𝑧 ∈ {𝑦𝜑}) ↔ (𝑧𝑥 ↔ [𝑧 / 𝑦]𝜑))
43albii 1822 . 2 (∀𝑧(𝑧𝑥𝑧 ∈ {𝑦𝜑}) ↔ ∀𝑧(𝑧𝑥 ↔ [𝑧 / 𝑦]𝜑))
51, 4bitri 278 1 (𝑥 = {𝑦𝜑} ↔ ∀𝑧(𝑧𝑥 ↔ [𝑧 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1537   = wceq 1539  [wsb 2070   ∈ wcel 2112  {cab 2736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-clab 2737  df-cleq 2751 This theorem is referenced by:  eliminable-abelv  34589  eliminable-abelab  34590
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