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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm14.12 | Structured version Visualization version GIF version |
Description: Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
pm14.12 | ⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2638 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | mo3 2623 | . . 3 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
4 | sbsbc 3724 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
5 | 4 | anbi2i 625 | . . . . 5 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
6 | 5 | imbi1i 353 | . . . 4 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
7 | 6 | 2albii 1822 | . . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
8 | 3, 7 | bitri 278 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
9 | 1, 8 | sylib 221 | 1 ⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 [wsb 2069 ∃*wmo 2596 ∃!weu 2628 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-sbc 3721 |
This theorem is referenced by: pm14.24 41136 |
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