Theorem disjeq1i 36388

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 36383	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
3	2	albii 1821	. 2 ⊢ (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
4		df-disj 5067	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶)
5		df-disj 5067	. 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 1540   = wceq 1542   ∈ wcel 2114  ∃*wrmo 3350  Disj wdisj 5066

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-mo 2540  df-cleq 2729  df-clel 2812  df-rmo 3351  df-disj 5067

This theorem is referenced by:  disjeq12i  36389