Theorem bj-nfcjust 33148

Description: Remove dependency on ax-ext 2732 (and df-cleq 2745 and ax-13 2383) from nfcjust 2882. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nfcjust	⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)

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

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bj-nfcjust

Step	Hyp	Ref	Expression
1		nfv 1984	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝑧
2		eleq1w 2814	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
3	1, 2	nfbidf 2231	. 2 ⊢ (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑧 ∈ 𝐴))
4	3	bj-cbvalvv 33031	1 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)

Syntax hints:   ↔ wb 196  ∀wal 1622  Ⅎwnf 1849   ∈ wcel 2131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-11 2175  ax-12 2188

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1846  df-nf 1851  df-clel 2748

This theorem is referenced by: (None)