Theorem disjeq12d 5078

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
disjeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
disjeq12d.1	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
disjeq12d	⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem disjeq12d

Step	Hyp	Ref	Expression
1		disjeq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	disjeq1d 5077	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
3		disjeq12d.1	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
5	4	disjeq2dv 5074	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))
6	2, 5	bitrd 281	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144  Disj wdisj 5069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-mo 2568  df-cleq 2756  df-clel 2839  df-ral 3079  df-rmo 3369  df-ss 3923  df-disj 5070

This theorem is referenced by: (None)