Theorem disjeq12d 4019

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 4018	. 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 4015	. 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 1364   ∈ wcel 2167  Disj wdisj 4010

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-rmo 2483  df-in 3163  df-ss 3170  df-disj 4011

This theorem is referenced by: (None)