Theorem nfpr 3774

Description: Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)

Hypotheses

Ref	Expression
nfpr.1	⊢ ℲxA
nfpr.2	⊢ ℲxB

Assertion

Ref	Expression
nfpr	⊢ Ⅎx{A, B}

Proof of Theorem nfpr

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfpr2 3750	. 2 ⊢ {A, B} = {y ∣ (y = A ∨ y = B)}
2		nfpr.1	. . . . 5 ⊢ ℲxA
3	2	nfeq2 2501	. . . 4 ⊢ Ⅎx y = A
4		nfpr.2	. . . . 5 ⊢ ℲxB
5	4	nfeq2 2501	. . . 4 ⊢ Ⅎx y = B
6	3, 5	nfor 1836	. . 3 ⊢ Ⅎx(y = A ∨ y = B)
7	6	nfab 2494	. 2 ⊢ Ⅎx{y ∣ (y = A ∨ y = B)}
8	1, 7	nfcxfr 2487	1 ⊢ Ⅎx{A, B}

Syntax hints:   ∨ wo 357   = wceq 1642  {cab 2339  Ⅎwnfc 2477  {cpr 3739

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-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743

This theorem is referenced by:  nfsn  3785  nfopk  4069