Theorem nfdisj1 5077

Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
nfdisj1	⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfdisj1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 5064	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		nfrmo1 3375	. . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfal 2326	. 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
4	1, 3	nfxfr 1854	1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:  ∀wal 1539  Ⅎwnf 1784   ∈ wcel 2113  ∃*wrmo 3347  Disj wdisj 5063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2146  ax-11 2162  ax-12 2182

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-mo 2537  df-rmo 3348  df-disj 5064

This theorem is referenced by:  disjabrex  32606  disjabrexf  32607  hasheuni  34191  ldgenpisyslem1  34269  measvunilem  34318  measvunilem0  34319  measvuni  34320  measinblem  34326  voliune  34335  volfiniune  34336  volmeas  34337  dstrvprob  34578  ismeannd  46653