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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege57c | Structured version Visualization version GIF version |
Description: Swap order of implication in ax-frege52c 40254. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege57c.a | ⊢ 𝐴 ∈ 𝐶 |
Ref | Expression |
---|---|
frege57c | ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege52c 40254 | . 2 ⊢ (𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) | |
2 | frege57c.a | . . 3 ⊢ 𝐴 ∈ 𝐶 | |
3 | 2 | frege56c 40285 | . 2 ⊢ ((𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) → (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-frege1 40156 ax-frege2 40157 ax-frege8 40175 ax-frege52c 40254 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3773 df-sn 4568 |
This theorem is referenced by: (None) |
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