Theorem cbvdisjv 5006

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 2955	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2955	. 2 ⊢ Ⅎ𝑥𝐶
3		cbvdisjv.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbvdisj 5005	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  Disj wdisj 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rmo 3114  df-disj 4996

This theorem is referenced by:  uniioombllem4  24190  hashunif  30554  tocyccntz  30836  totprob  31795  disjrnmpt2  41815  ismeannd  43106  psmeasure  43110  volmea  43113  meaiuninclem  43119  caratheodorylem1  43165  caratheodory  43167