Theorem bnj1113 34981

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1113.1	⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)

Assertion

Ref	Expression
bnj1113	⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸)

Distinct variable groups:   𝑥,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐸(𝑥)

Proof of Theorem bnj1113

Step	Hyp	Ref	Expression
1		bnj1113.1	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷)
2	1	iuneq1d 4951	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸)

Syntax hints:   → wi 4   = wceq 1548  ∪ ciun 4923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rex 3066  df-v 3435  df-ss 3901  df-iun 4925

This theorem is referenced by:  bnj106  35063  bnj222  35078  bnj540  35087  bnj553  35093  bnj611  35113  bnj966  35139  bnj1112  35178