Theorem iuneq12f 39235

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 4973	. 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 39228	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷)
5	1, 4	sylan9eqr 2820	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  Ⅎwnfc 2910  ∀wral 3079  ∪ ciun 4953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-v 3455  df-ss 3900  df-iun 4955

This theorem is used by: (None)