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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 3771 | . . 3 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵}) | |
2 | eqeq1 2823 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | 2 | elabg 3664 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} → (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} ↔ 𝐴 = 𝐵)) |
4 | 3 | ibi 269 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} → 𝐴 = 𝐵) |
5 | 1, 4 | sylbi 219 | . 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 1531 ∈ wcel 2108 {cab 2797 [wsbc 3770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-sbc 3771 |
This theorem is referenced by: frege56c 40255 |
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