Theorem disjeq1f 31733

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

Hypotheses

Ref	Expression
disjss1f.1	⊢ Ⅎ𝑥𝐴
disjss1f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
disjeq1f	⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))

Proof of Theorem disjeq1f

Step	Hyp	Ref	Expression
1		eqimss2 4038	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		disjss1f.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3		disjss1f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	disjss1f 31732	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐵 𝐶))
5	1, 4	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐵 𝐶))
6		eqimss 4037	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7	3, 2	disjss1f 31732	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))
8	6, 7	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))
9	5, 8	impbid 211	1 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541  Ⅎwnfc 2883   ⊆ wss 3945  Disj wdisj 5107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rmo 3376  df-v 3476  df-in 3952  df-ss 3962  df-disj 5108

This theorem is referenced by:  ldgenpisyslem1  33056