Theorem cbvdisjf 30811

Description: Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)

Hypotheses

Ref	Expression
cbvdisjf.1	⊢ Ⅎ𝑥𝐴
cbvdisjf.2	⊢ Ⅎ𝑦𝐵
cbvdisjf.3	⊢ Ⅎ𝑥𝐶
cbvdisjf.4	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem cbvdisjf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
2		cbvdisjf.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
3	2	nfcri 2893	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
4	1, 3	nfan 1903	. . . . 5 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
5		cbvdisjf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
6	5	nfcri 2893	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
7		cbvdisjf.3	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
8	7	nfcri 2893	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
9	6, 8	nfan 1903	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)
10		eleq1w 2821	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
11		cbvdisjf.4	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
12	11	eleq2d 2824	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
13	10, 12	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
14	4, 9, 13	cbvmow 2603	. . . 4 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))
15		df-rmo 3071	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
16		df-rmo 3071	. . . 4 ⊢ (∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))
17	14, 15, 16	3bitr4i 302	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
18	17	albii 1823	. 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
19		df-disj 5036	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
20		df-disj 5036	. 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
21	18, 19, 20	3bitr4i 302	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  Ⅎwnfc 2886  ∃*wrmo 3066  Disj wdisj 5035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-mo 2540  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rmo 3071  df-disj 5036

This theorem is referenced by:  disjorsf  30820  ldgenpisyslem1  32031