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Mirrors > Home > MPE Home > Th. List > sbmo | Structured version Visualization version GIF version |
Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
sbmo | ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbex 2491 | . . 3 ⊢ ([𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤)) | |
2 | nfv 1883 | . . . . . 6 ⊢ Ⅎ𝑥 𝑧 = 𝑤 | |
3 | 2 | sblim 2425 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤)) |
4 | 3 | sbalv 2492 | . . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤)) |
5 | 4 | exbii 1814 | . . 3 ⊢ (∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤)) |
6 | 1, 5 | bitri 264 | . 2 ⊢ ([𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤)) |
7 | mo2v 2505 | . . 3 ⊢ (∃*𝑧𝜑 ↔ ∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤)) | |
8 | 7 | sbbii 1944 | . 2 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤)) |
9 | mo2v 2505 | . 2 ⊢ (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤)) | |
10 | 6, 8, 9 | 3bitr4i 292 | 1 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1521 ∃wex 1744 [wsb 1937 ∃*wmo 2499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 |
This theorem is referenced by: (None) |
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