Theorem eliminable2a 37461

Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
eliminable2a	⊢ (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 eliminable2a

Step	Hyp	Ref	Expression
1		dfcleq 2754	1 ⊢ (𝑥 = {𝑦 ∣ 𝜑} ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑦 ∣ 𝜑}))

Syntax hints:   ↔ wb 209  ∀wal 1566   = wceq 1568   ∈ wcel 2141  {cab 2739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-cleq 2753

This theorem is referenced by: (None)