Theorem bj-nfsab1 37471

Description: Remove dependency on ax-13 2404 from nfsab1 2749. UPDATE / TODO: nfsab1 2749 does not use ax-13 2404 either anymore; bj-nfsab1 37471 is shorter than nfsab1 2749 but uses ax-12 2213. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nfsab1	⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-nfsab1

Step	Hyp	Ref	Expression
1		hbab1 2750	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
2	1	nf5i 2181	1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1813   ∈ wcel 2143  {cab 2741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742

This theorem is referenced by: (None)