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| Description: Lemma for frege57c 43933. 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 43928 | . . . 4 ⊢ [𝐵 / 𝑥]𝑥 = 𝐵 | 
| 3 | frege53c 43927 | . . . 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 43929 | . . 3 ⊢ ((𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴 → 𝐴 = 𝐵)) | |
| 6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐵 = 𝐴 → 𝐴 = 𝐵) | 
| 7 | frege9 43825 | . 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 1540 ∈ wcel 2108 [wsbc 3788 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-frege1 43803 ax-frege2 43804 ax-frege8 43822 ax-frege52c 43901 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 df-sn 4627 | 
| This theorem is referenced by: frege57c 43933 | 
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