Theorem disjeq12i 36738

Description: Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
disjeq12i.1	⊢ 𝐴 = 𝐵
disjeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
disjeq12i	⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)

Proof of Theorem disjeq12i

Step	Hyp	Ref	Expression
1		disjeq2 5085	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐷))
2		disjeq12i.2	. . . 4 ⊢ 𝐶 = 𝐷
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐶 = 𝐷)
4	1, 3	mprg 3088	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐷)
5		disjeq12i.1	. . 3 ⊢ 𝐴 = 𝐵
6	5	disjeq1i 36737	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐷 ↔ Disj 𝑥 ∈ 𝐵 𝐷)
7	4, 6	bitri 278	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2146  Disj wdisj 5081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2570  df-cleq 2758  df-clel 2841  df-ral 3083  df-rmo 3372  df-ss 3925  df-disj 5082

This theorem is used by: (None)