Theorem bnj1113 34799

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 5019	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸)

Syntax hints:   → wi 4   = wceq 1540  ∪ ciun 4991

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-ss 3968  df-iun 4993

This theorem is referenced by:  bnj106  34882  bnj222  34897  bnj540  34906  bnj553  34912  bnj611  34932  bnj966  34958  bnj1112  34997