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Theorem frege55lem1c 39768
 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 3712 . . 3 ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 ∈ {𝑥𝑥 = 𝐵})
2 eqeq1 2801 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
32elabg 3607 . . . 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 1525   ∈ wcel 2083  {cab 2777  [wsbc 3711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-sbc 3712 This theorem is referenced by:  frege56c  39771
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