Theorem bnj956

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

	Ref	Expression
Hypothesis	bnj956.1	⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 𝐴 = 𝐵 )
Assertion	bnj956	⊢ ( 𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		bnj956.1	⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 𝐴 = 𝐵 )
2		eleq2	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
3	2	anbi1d	⊢ ( 𝐴 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )
4	3	alexbii	⊢ ( ∀ 𝑥 𝐴 = 𝐵 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )
5		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) )
6		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) )
7	4 5 6	3bitr4g	⊢ ( ∀ 𝑥 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ) )
8	1 7	syl	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ) )
9	8	abbidv	⊢ ( 𝐴 = 𝐵 → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 } )
10		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 }
11		df-iun	⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 }
12	9 10 11	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )