Theorem iuneq2df 44034

Description: Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
iuneq2df.1	⊢ Ⅎ𝑥𝜑
iuneq2df.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
iuneq2df	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Proof of Theorem iuneq2df

Step	Hyp	Ref	Expression
1		iuneq2df.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		iuneq2df.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	ex 411	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶))
4	1, 3	ralrimi 3252	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
5		iuneq2 5015	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
6	4, 5	syl 17	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539  Ⅎwnf 1783   ∈ wcel 2104  ∀wral 3059  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-v 3474  df-in 3954  df-ss 3964  df-iun 4998

This theorem is referenced by:  subsaliuncl  45372  omeiunlempt  45534  hoicvrrex  45570  ovnlecvr2  45624  smflimmpt  45824  smflimsupmpt  45843  smfliminfmpt  45846