Theorem iuneq12f 33972

Description: Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Hypotheses

Ref	Expression
iuneq12f.1	⊢ Ⅎ𝑥𝐴
iuneq12f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
iuneq12f	⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Proof of Theorem iuneq12f

Step	Hyp	Ref	Expression
1		iuneq2 4537	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐴 𝐷)
2		iuneq12f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		iuneq12f.2	. . 3 ⊢ Ⅎ𝑥𝐵
4	2, 3	iuneq2f 33963	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷)
5	1, 4	sylan9eqr 2678	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  Ⅎwnfc 2751  ∀wral 2912  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by: (None)