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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege55lem1c | Structured version Visualization version GIF version |
Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
frege55lem1c | ⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3712 | . . 3 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵}) | |
2 | eqeq1 2801 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | 2 | elabg 3607 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} → (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} ↔ 𝐴 = 𝐵)) |
4 | 3 | ibi 268 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} → 𝐴 = 𝐵) |
5 | 1, 4 | sylbi 218 | . 2 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐵 → 𝐴 = 𝐵) |
6 | 5 | imim2i 16 | 1 ⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 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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