Theorem cbvdisj 4630

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 2758	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbvdisj.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2758	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
5		cbvdisj.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eleq2d 2687	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7	2, 4, 6	cbvrmo 3170	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	albii 1747	. 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
9		df-disj 4621	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
10		df-disj 4621	. 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
11	8, 9, 10	3bitr4i 292	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751  ∃*wrmo 2915  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-disj 4621

This theorem is referenced by:  cbvdisjv  4631  disjors  4635  disjxiun  4649  disjxiunOLD  4650  volfiniun  23315  voliun  23322  carsggect  30380  omsmeas  30385  disjf1  39369  disjrnmpt2  39375  fsumiunss  39807  sge0iunmpt  40635  iundjiun  40677  meadjiun  40683