Theorem iuneq12df 5023

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 2825	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
7	1, 2, 3, 4, 6	rexeqbid 3355	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
8	7	alrimiv 1925	. 2 ⊢ (𝜑 → ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
9		abbi 2805	. . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷})
10		df-iun 4998	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
11		df-iun 4998	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}
12	9, 10, 11	3eqtr4g 2800	. 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 206  ∀wal 1535   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106  {cab 2712  Ⅎwnfc 2888  ∃wrex 3068  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-iun 4998

This theorem is referenced by:  iunxdif3  5100  iundisjf  32609  aciunf1  32680  measvuni  34195  iuneq2f  38143