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Theorem subtr 32433
 Description: Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
subtr.1 𝑥𝐴
subtr.2 𝑥𝐵
subtr.3 𝑥𝑌
subtr.4 𝑥𝑍
subtr.5 (𝑥 = 𝐴𝑋 = 𝑌)
subtr.6 (𝑥 = 𝐵𝑋 = 𝑍)
Assertion
Ref Expression
subtr ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵𝑌 = 𝑍))

Proof of Theorem subtr
StepHypRef Expression
1 subtr.1 . . 3 𝑥𝐴
2 subtr.2 . . . . 5 𝑥𝐵
31, 2nfeq 2805 . . . 4 𝑥 𝐴 = 𝐵
4 subtr.3 . . . . 5 𝑥𝑌
5 subtr.4 . . . . 5 𝑥𝑍
64, 5nfeq 2805 . . . 4 𝑥 𝑌 = 𝑍
73, 6nfim 1865 . . 3 𝑥(𝐴 = 𝐵𝑌 = 𝑍)
8 eqeq1 2655 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
9 subtr.5 . . . . 5 (𝑥 = 𝐴𝑋 = 𝑌)
109eqeq1d 2653 . . . 4 (𝑥 = 𝐴 → (𝑋 = 𝑍𝑌 = 𝑍))
118, 10imbi12d 333 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐵𝑋 = 𝑍) ↔ (𝐴 = 𝐵𝑌 = 𝑍)))
12 subtr.6 . . 3 (𝑥 = 𝐵𝑋 = 𝑍)
131, 7, 11, 12vtoclgf 3295 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝑌 = 𝑍))
1413adantr 480 1 ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵𝑌 = 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Ⅎwnfc 2780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233 This theorem is referenced by: (None)
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