Theorem iuneq2d 3673

Description: Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)

Hypothesis

Ref	Expression
iuneq2d.2	⊢ (φ → B = 𝐶)

Assertion

Ref	Expression
iuneq2d	⊢ (φ → ∪ x ∈ A B = ∪ x ∈ A 𝐶)

Distinct variable groups:   φ,x   x,A

Allowed substitution hints:   B(x)   𝐶(x)

Proof of Theorem iuneq2d

Step	Hyp	Ref	Expression
1		iuneq2d.2	. . 3 ⊢ (φ → B = 𝐶)
2	1	adantr 261	. 2 ⊢ ((φ ∧ x ∈ A) → B = 𝐶)
3	2	iuneq2dv 3669	1 ⊢ (φ → ∪ x ∈ A B = ∪ x ∈ A 𝐶)

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∪ ciun 3648

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650

This theorem is referenced by:  rdgeq1  5898