Theorem disjeq1i 36652

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 36647	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
3	2	albii 1847	. 2 ⊢ (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
4		df-disj 5082	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶)
5		df-disj 5082	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
6	3, 4, 5	3bitr4i 306	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   ↔ wb 209  ∀wal 1566   = wceq 1568   ∈ wcel 2150  ∃*wrmo 3375  Disj wdisj 5081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-mo 2574  df-cleq 2762  df-clel 2845  df-rmo 3376  df-disj 5082

This theorem is referenced by:  disjeq12i  36653