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Theorem iuneq12d 3672
Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
Hypotheses
Ref Expression
iuneq1d.1 (φA = B)
iuneq12d.2 (φ𝐶 = 𝐷)
Assertion
Ref Expression
iuneq12d (φ x A 𝐶 = x B 𝐷)
Distinct variable groups:   x,A   x,B   φ,x
Allowed substitution hints:   𝐶(x)   𝐷(x)

Proof of Theorem iuneq12d
StepHypRef Expression
1 iuneq1d.1 . . 3 (φA = B)
21iuneq1d 3671 . 2 (φ x A 𝐶 = x B 𝐶)
3 iuneq12d.2 . . . 4 (φ𝐶 = 𝐷)
43adantr 261 . . 3 ((φ x B) → 𝐶 = 𝐷)
54iuneq2dv 3669 . 2 (φ x B 𝐶 = x B 𝐷)
62, 5eqtrd 2069 1 (φ x A 𝐶 = x B 𝐷)
Colors of variables: wff set class
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-bnd 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:  rdgivallem  5908  rdg0  5914
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