Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ssbn | Structured version Visualization version GIF version |
Description: The result of a substitution in the negation of a formula is the negation of the result of the same substitution in that formula. Proved from Tarski, ax-10 2006, bj-ax12 31823. Compare sbn 2379. (Contributed by BJ, 25-Dec-2020.) |
Ref | Expression |
---|---|
bj-ssbn | ⊢ ([𝑡/𝑥]b ¬ 𝜑 ↔ ¬ [𝑡/𝑥]b𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ssb 31809 | . 2 ⊢ ([𝑡/𝑥]b ¬ 𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))) | |
2 | alinexa 1759 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
3 | 2 | imbi2i 325 | . . 3 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
4 | 3 | albii 1737 | . 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
5 | alinexa 1759 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
6 | bj-dfssb2 31829 | . . 3 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
7 | 5, 6 | xchbinxr 324 | . 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ [𝑡/𝑥]b𝜑) |
8 | 1, 4, 7 | 3bitri 285 | 1 ⊢ ([𝑡/𝑥]b ¬ 𝜑 ↔ ¬ [𝑡/𝑥]b𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 ∃wex 1695 [wssb 31808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-ssb 31809 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |