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Theorem frege55lem1c 40140
Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
Assertion
Ref Expression
frege55lem1c ((𝜑[𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑𝐴 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem frege55lem1c
StepHypRef Expression
1 df-sbc 3770 . . 3 ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 ∈ {𝑥𝑥 = 𝐵})
2 eqeq1 2822 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
32elabg 3663 . . . 4 (𝐴 ∈ {𝑥𝑥 = 𝐵} → (𝐴 ∈ {𝑥𝑥 = 𝐵} ↔ 𝐴 = 𝐵))
43ibi 268 . . 3 (𝐴 ∈ {𝑥𝑥 = 𝐵} → 𝐴 = 𝐵)
51, 4sylbi 218 . 2 ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵)
65imim2i 16 1 ((𝜑[𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {cab 2796  [wsbc 3769
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-sbc 3770
This theorem is referenced by:  frege56c  40143
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