Theorem disjeq2 3723

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjeq2	⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A B ↔ Disj x ∈ A 𝐶))

Proof of Theorem disjeq2

Step	Hyp	Ref	Expression
1		eqimss2 2975	. . . 4 ⊢ (B = 𝐶 → 𝐶 ⊆ B)
2	1	ralimi 2362	. . 3 ⊢ (∀x ∈ A B = 𝐶 → ∀x ∈ A 𝐶 ⊆ B)
3		disjss2 3722	. . 3 ⊢ (∀x ∈ A 𝐶 ⊆ B → (Disj x ∈ A B → Disj x ∈ A 𝐶))
4	2, 3	syl 14	. 2 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A B → Disj x ∈ A 𝐶))
5		eqimss 2974	. . . 4 ⊢ (B = 𝐶 → B ⊆ 𝐶)
6	5	ralimi 2362	. . 3 ⊢ (∀x ∈ A B = 𝐶 → ∀x ∈ A B ⊆ 𝐶)
7		disjss2 3722	. . 3 ⊢ (∀x ∈ A B ⊆ 𝐶 → (Disj x ∈ A 𝐶 → Disj x ∈ A B))
8	6, 7	syl 14	. 2 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A 𝐶 → Disj x ∈ A B))
9	4, 8	impbid 120	1 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A B ↔ Disj x ∈ A 𝐶))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228  ∀wral 2284   ⊆ wss 2894  Disj wdisj 3719

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-ral 2289  df-rmo 2292  df-in 2901  df-ss 2908  df-disj 3720

This theorem is referenced by:  disjeq2dv  3724