![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege56c | Structured version Visualization version GIF version |
Description: Lemma for frege57c 43126. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege56c.b | ⊢ 𝐵 ∈ 𝐶 |
Ref | Expression |
---|---|
frege56c | ⊢ ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege56c.b | . . . . 5 ⊢ 𝐵 ∈ 𝐶 | |
2 | 1 | frege54cor1c 43121 | . . . 4 ⊢ [𝐵 / 𝑥]𝑥 = 𝐵 |
3 | frege53c 43120 | . . . 4 ⊢ ([𝐵 / 𝑥]𝑥 = 𝐵 → (𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) |
5 | frege55lem1c 43122 | . . 3 ⊢ ((𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴 → 𝐴 = 𝐵)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐵 = 𝐴 → 𝐴 = 𝐵) |
7 | frege9 43018 | . 2 ⊢ ((𝐵 = 𝐴 → 𝐴 = 𝐵) → ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)))) | |
8 | 6, 7 | ax-mp 5 | 1 ⊢ ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-frege1 42996 ax-frege2 42997 ax-frege8 43015 ax-frege52c 43094 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-sbc 3770 df-sn 4621 |
This theorem is referenced by: frege57c 43126 |
Copyright terms: Public domain | W3C validator |