Theorem disjeq1d 5099

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

Hypothesis

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

Assertion

Ref	Expression
disjeq1d	⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem disjeq1d

Step	Hyp	Ref	Expression
1		disjeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		disjeq1 5098	. 2 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
3	1, 2	syl 17	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  Disj wdisj 5091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2540  df-cleq 2728  df-clel 2810  df-rmo 3364  df-ss 3948  df-disj 5092

This theorem is referenced by:  disjeq12d  5100  disjxiun  5121  disjdifprg  32561  disjdifprg2  32562  disjun0  32581  tocyccntz  33160  measxun2  34246  measssd  34251  meadjun  46458