Theorem bnj1113 35183

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 4983	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∪ ciun 4955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3456  df-ss 3921  df-iun 4957

This theorem is used by:  bnj106  35265  bnj222  35280  bnj540  35289  bnj553  35295  bnj611  35315  bnj966  35341  bnj1112  35380