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Theorem subtr2 33560
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 2988 . . . 4 𝑥 𝐴 = 𝐵
4 subtr2.3 . . . . 5 𝑥𝜓
5 subtr2.4 . . . . 5 𝑥𝜒
64, 5nfbi 1895 . . . 4 𝑥(𝜓𝜒)
73, 6nfim 1888 . . 3 𝑥(𝐴 = 𝐵 → (𝜓𝜒))
8 eqeq1 2822 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
9 subtr2.5 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
109bibi1d 345 . . . 4 (𝑥 = 𝐴 → ((𝜑𝜒) ↔ (𝜓𝜒)))
118, 10imbi12d 346 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐵 → (𝜑𝜒)) ↔ (𝐴 = 𝐵 → (𝜓𝜒))))
12 subtr2.6 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
131, 7, 11, 12vtoclgf 3563 . 2 (𝐴𝐶 → (𝐴 = 𝐵 → (𝜓𝜒)))
1413adantr 481 1 ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵 → (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wnf 1775  wcel 2105  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494
This theorem is referenced by: (None)
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