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Theorem subtr 31947
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 2772 . . . 4 𝑥 𝐴 = 𝐵
4 subtr.3 . . . . 5 𝑥𝑌
5 subtr.4 . . . . 5 𝑥𝑍
64, 5nfeq 2772 . . . 4 𝑥 𝑌 = 𝑍
73, 6nfim 1822 . . 3 𝑥(𝐴 = 𝐵𝑌 = 𝑍)
8 eqeq1 2625 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
9 subtr.5 . . . . 5 (𝑥 = 𝐴𝑋 = 𝑌)
109eqeq1d 2623 . . . 4 (𝑥 = 𝐴 → (𝑋 = 𝑍𝑌 = 𝑍))
118, 10imbi12d 334 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐵𝑋 = 𝑍) ↔ (𝐴 = 𝐵𝑌 = 𝑍)))
12 subtr.6 . . 3 (𝑥 = 𝐵𝑋 = 𝑍)
131, 7, 11, 12vtoclgf 3250 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝑌 = 𝑍))
1413adantr 481 1 ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵𝑌 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wnfc 2748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188
This theorem is referenced by: (None)
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