Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > pm13.192 | Structured version Visualization version GIF version |
Description: Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
Ref | Expression |
---|---|
pm13.192 | ⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimpr 222 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝑥 = 𝐴)) | |
2 | 1 | alimi 1812 | . . . . . 6 ⊢ (∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝐴)) |
3 | eqeq1 2827 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
4 | 3 | equsalvw 2010 | . . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝐴) ↔ 𝑦 = 𝐴) |
5 | 2, 4 | sylib 220 | . . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) → 𝑦 = 𝐴) |
6 | eqeq2 2835 | . . . . . . 7 ⊢ (𝐴 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑥 = 𝑦)) | |
7 | 6 | eqcoms 2831 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝐴 ↔ 𝑥 = 𝑦)) |
8 | 7 | alrimiv 1928 | . . . . 5 ⊢ (𝑦 = 𝐴 → ∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦)) |
9 | 5, 8 | impbii 211 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ↔ 𝑦 = 𝐴) |
10 | 9 | anbi1i 625 | . . 3 ⊢ ((∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ (𝑦 = 𝐴 ∧ 𝜑)) |
11 | 10 | exbii 1848 | . 2 ⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) |
12 | sbc5 3802 | . 2 ⊢ ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) | |
13 | 11, 12 | bitr4i 280 | 1 ⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∃wex 1780 [wsbc 3774 |
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-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-v 3498 df-sbc 3775 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |