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Theorem frege55lem1c 40991
 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 3698 . . 3 ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 ∈ {𝑥𝑥 = 𝐵})
2 eqeq1 2763 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
32elabg 3588 . . . 4 (𝐴 ∈ {𝑥𝑥 = 𝐵} → (𝐴 ∈ {𝑥𝑥 = 𝐵} ↔ 𝐴 = 𝐵))
43ibi 270 . . 3 (𝐴 ∈ {𝑥𝑥 = 𝐵} → 𝐴 = 𝐵)
51, 4sylbi 220 . 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 1539   ∈ wcel 2112  {cab 2736  [wsbc 3697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-sbc 3698 This theorem is referenced by:  frege56c  40994
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