Theorem iuneq2df 45889

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 418	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶))
4	1, 3	ralrimi 3262	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
5		iuneq2 4974	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
6	4, 5	syl 18	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  ∀wral 3078  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-v 3455  df-ss 3919  df-iun 4956

This theorem is used by:  subsaliuncl  47194  omeiunlempt  47356  hoicvrrex  47392  ovnlecvr2  47446  smflimmpt  47646  smflimsupmpt  47665  smfliminfmpt  47668