Theorem iuneq12df 4945

Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)

Hypotheses

Ref	Expression
iuneq12df.1	⊢ Ⅎ𝑥𝜑
iuneq12df.2	⊢ Ⅎ𝑥𝐴
iuneq12df.3	⊢ Ⅎ𝑥𝐵
iuneq12df.4	⊢ (𝜑 → 𝐴 = 𝐵)
iuneq12df.5	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
iuneq12df	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Proof of Theorem iuneq12df

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq12df.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		iuneq12df.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		iuneq12df.3	. . . 4 ⊢ Ⅎ𝑥𝐵
4		iuneq12df.4	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
5		iuneq12df.5	. . . . 5 ⊢ (𝜑 → 𝐶 = 𝐷)
6	5	eleq2d 2898	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
7	1, 2, 3, 4, 6	rexeqbid 3422	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
8	7	alrimiv 1928	. 2 ⊢ (𝜑 → ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
9		abbi1 2884	. . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷})
10		df-iun 4921	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
11		df-iun 4921	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}
12	9, 10, 11	3eqtr4g 2881	. 2 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
13	8, 12	syl 17	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  {cab 2799  Ⅎwnfc 2961  ∃wrex 3139  ∪ ciun 4919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-iun 4921

This theorem is referenced by:  iunxdif3  5017  iundisjf  30339  aciunf1  30408  measvuni  31473  iuneq2f  35449