Theorem disjeq12d 5048

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 5047	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
3		disjeq12d.1	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
5	4	disjeq2dv 5044	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))
6	2, 5	bitrd 278	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Disj wdisj 5039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rmo 3071  df-v 3434  df-in 3894  df-ss 3904  df-disj 5040

This theorem is referenced by: (None)