Theorem cbvdisjv 5119

Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
cbvdisjv.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvdisjv	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvdisjv

Step	Hyp	Ref	Expression
1		nfcv 2899	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2899	. 2 ⊢ Ⅎ𝑥𝐶
3		cbvdisjv.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbvdisj 5118	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534  Disj wdisj 5108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2530  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rmo 3372  df-disj 5109

This theorem is referenced by:  uniioombllem4  25509  hashunif  32570  tocyccntz  32860  totprob  34042  disjrnmpt2  44552  ismeannd  45846  psmeasure  45850  volmea  45853  meaiuninclem  45859  caratheodorylem1  45905  caratheodory  45907