Theorem disjeq12d 4078

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 4077	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
3		disjeq12d.1	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
5	4	disjeq2dv 4074	. 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))
6	2, 5	bitrd 188	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2202  Disj wdisj 4069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2516  df-rmo 2519  df-in 3207  df-ss 3214  df-disj 4070

This theorem is referenced by: (None)