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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege56c | Structured version Visualization version GIF version |
Description: Lemma for frege57c 41074. 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 41069 | . . . 4 ⊢ [𝐵 / 𝑥]𝑥 = 𝐵 |
3 | frege53c 41068 | . . . 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 41070 | . . 3 ⊢ ((𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴 → 𝐴 = 𝐵)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐵 = 𝐴 → 𝐴 = 𝐵) |
7 | frege9 40966 | . 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 1542 ∈ wcel 2114 [wsbc 3680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-frege1 40944 ax-frege2 40945 ax-frege8 40963 ax-frege52c 41042 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-sbc 3681 df-sn 4517 |
This theorem is referenced by: frege57c 41074 |
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