Theorem disjeq12i 36172

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 5112	. . 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 3066	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐷)
5		disjeq12i.1	. . 3 ⊢ 𝐴 = 𝐵
6	5	disjeq1i 36171	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐷 ↔ Disj 𝑥 ∈ 𝐵 𝐷)
7	4, 6	bitri 275	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2108  Disj wdisj 5108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2539  df-cleq 2728  df-clel 2815  df-ral 3061  df-rmo 3379  df-ss 3967  df-disj 5109

This theorem is referenced by: (None)