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Theorem subtr2 33776
Description: Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
subtr.1 𝑥𝐴
subtr.2 𝑥𝐵
subtr2.3 𝑥𝜓
subtr2.4 𝑥𝜒
subtr2.5 (𝑥 = 𝐴 → (𝜑𝜓))
subtr2.6 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
subtr2 ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵 → (𝜓𝜒)))

Proof of Theorem subtr2
StepHypRef Expression
1 subtr.1 . . 3 𝑥𝐴
2 subtr.2 . . . . 5 𝑥𝐵
31, 2nfeq 2968 . . . 4 𝑥 𝐴 = 𝐵
4 subtr2.3 . . . . 5 𝑥𝜓
5 subtr2.4 . . . . 5 𝑥𝜒
64, 5nfbi 1904 . . . 4 𝑥(𝜓𝜒)
73, 6nfim 1897 . . 3 𝑥(𝐴 = 𝐵 → (𝜓𝜒))
8 eqeq1 2802 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
9 subtr2.5 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
109bibi1d 347 . . . 4 (𝑥 = 𝐴 → ((𝜑𝜒) ↔ (𝜓𝜒)))
118, 10imbi12d 348 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐵 → (𝜑𝜒)) ↔ (𝐴 = 𝐵 → (𝜓𝜒))))
12 subtr2.6 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
131, 7, 11, 12vtoclgf 3513 . 2 (𝐴𝐶 → (𝐴 = 𝐵 → (𝜓𝜒)))
1413adantr 484 1 ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵 → (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2111  wnfc 2936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443
This theorem is referenced by: (None)
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