Theorem disjeq1i 36148

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

Hypothesis

Ref	Expression
disjeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
disjeq1i	⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)

Proof of Theorem disjeq1i

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjeq1i.1	. . . 4 ⊢ 𝐴 = 𝐵
2	1	rmoeqi 36143	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
3	2	albii 1817	. 2 ⊢ (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
4		df-disj 5134	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶)
5		df-disj 5134	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
6	3, 4, 5	3bitr4i 303	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∃*wrmo 3387  Disj wdisj 5133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-mo 2543  df-cleq 2732  df-clel 2819  df-rmo 3388  df-disj 5134

This theorem is referenced by:  disjeq12i  36149