Theorem nfcjust 2739

Description: Justification theorem for df-nfc 2740. (Contributed by Mario Carneiro, 13-Oct-2016.)

Assertion

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

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

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfcjust

Step	Hyp	Ref	Expression
1		nfv 1830	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝑧
2		eleq1 2676	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
3	1, 2	nfbidf 2079	. 2 ⊢ (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑧 ∈ 𝐴))
4	3	cbvalv 2261	1 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)

Syntax hints:   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606

This theorem is referenced by: (None)