Theorem nfcjust 2477

Description: Justification theorem for df-nfc 2478. (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 1619	. . 3 ⊢ Ⅎx y = z
2		eleq1 2413	. . 3 ⊢ (y = z → (y ∈ A ↔ z ∈ A))
3	1, 2	nfbidf 1774	. 2 ⊢ (y = z → (Ⅎx y ∈ A ↔ Ⅎx z ∈ A))
4	3	cbvalv 2002	1 ⊢ (∀yℲx y ∈ A ↔ ∀zℲx z ∈ A)

Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349

This theorem is referenced by: (None)