Theorem cbvdisj 5068

Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
cbvdisj.1	⊢ Ⅎ𝑦𝐵
cbvdisj.2	⊢ Ⅎ𝑥𝐶
cbvdisj.3	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

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

Proof of Theorem cbvdisj

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvdisj.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
2	1	nfcri 2886	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbvdisj.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2886	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
5		cbvdisj.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eleq2d 2817	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7	2, 4, 6	cbvrmow 3371	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	albii 1820	. 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
9		df-disj 5059	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
10		df-disj 5059	. 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
11	8, 9, 10	3bitr4i 303	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879  ∃*wrmo 3345  Disj wdisj 5058

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-8 2113  ax-9 2121  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-mo 2535  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rmo 3346  df-disj 5059

This theorem is referenced by:  disjors  5074  disjxiun  5088  volfiniun  25476  voliun  25483  carsggect  34329  omsmeas  34334  disjf1  45226  disjrnmpt2  45231  fsumiunss  45621  sge0iunmpt  46462  iundjiun  46504  meadjiun  46510