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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege56c | Structured version Visualization version GIF version |
Description: Lemma for frege57c 43910. 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 43905 | . . . 4 ⊢ [𝐵 / 𝑥]𝑥 = 𝐵 |
3 | frege53c 43904 | . . . 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 43906 | . . 3 ⊢ ((𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴 → 𝐴 = 𝐵)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐵 = 𝐴 → 𝐴 = 𝐵) |
7 | frege9 43802 | . 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 1537 ∈ wcel 2106 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-frege1 43780 ax-frege2 43781 ax-frege8 43799 ax-frege52c 43878 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-sbc 3792 df-sn 4632 |
This theorem is referenced by: frege57c 43910 |
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