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Theorem sbceq1ddi 35473
Description: A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypotheses
Ref Expression
sbceq1ddi.1 (𝜑𝐴 = 𝐵)
sbceq1ddi.2 (𝜓𝜃)
sbceq1ddi.3 ([𝐴 / 𝑥]𝜒𝜃)
sbceq1ddi.4 ([𝐵 / 𝑥]𝜒𝜂)
Assertion
Ref Expression
sbceq1ddi ((𝜑𝜓) → 𝜂)

Proof of Theorem sbceq1ddi
StepHypRef Expression
1 sbceq1ddi.1 . . . 4 (𝜑𝐴 = 𝐵)
21adantr 484 . . 3 ((𝜑𝜓) → 𝐴 = 𝐵)
3 sbceq1ddi.2 . . . . 5 (𝜓𝜃)
4 sbceq1ddi.3 . . . . 5 ([𝐴 / 𝑥]𝜒𝜃)
53, 4sylibr 237 . . . 4 (𝜓[𝐴 / 𝑥]𝜒)
65adantl 485 . . 3 ((𝜑𝜓) → [𝐴 / 𝑥]𝜒)
72, 6sbceq1dd 3764 . 2 ((𝜑𝜓) → [𝐵 / 𝑥]𝜒)
8 sbceq1ddi.4 . 2 ([𝐵 / 𝑥]𝜒𝜂)
97, 8sylib 221 1 ((𝜑𝜓) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  [wsbc 3758
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896  df-sbc 3759
This theorem is referenced by: (None)
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