Theorem cbvdisj 5122

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 2888	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbvdisj.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2888	. . . 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 3403	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	albii 1819	. 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
9		df-disj 5113	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
10		df-disj 5113	. 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
11	8, 9, 10	3bitr4i 302	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2104  Ⅎwnfc 2881  ∃*wrmo 3373  Disj wdisj 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-mo 2532  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rmo 3374  df-disj 5113

This theorem is referenced by:  cbvdisjv  5123  disjors  5128  disjxiun  5144  volfiniun  25296  voliun  25303  carsggect  33615  omsmeas  33620  disjf1  44180  disjrnmpt2  44185  fsumiunss  44589  sge0iunmpt  45432  iundjiun  45474  meadjiun  45480