Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-moteq | Structured version Visualization version GIF version |
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.) |
Ref | Expression |
---|---|
wl-moteq | ⊢ (∃*𝑥⊤ → 𝑦 = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2622 | . 2 ⊢ (∃*𝑥⊤ ↔ ∃𝑤∀𝑥(⊤ → 𝑥 = 𝑤)) | |
2 | stdpc5v 1939 | . . . 4 ⊢ (∀𝑥(⊤ → 𝑥 = 𝑤) → (⊤ → ∀𝑥 𝑥 = 𝑤)) | |
3 | tru 1541 | . . . . . 6 ⊢ ⊤ | |
4 | 3 | pm2.24i 153 | . . . . 5 ⊢ (¬ ⊤ → 𝑦 = 𝑧) |
5 | aeveq 2061 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑤 → 𝑦 = 𝑧) | |
6 | 4, 5 | ja 188 | . . . 4 ⊢ ((⊤ → ∀𝑥 𝑥 = 𝑤) → 𝑦 = 𝑧) |
7 | 2, 6 | syl 17 | . . 3 ⊢ (∀𝑥(⊤ → 𝑥 = 𝑤) → 𝑦 = 𝑧) |
8 | 7 | exlimiv 1931 | . 2 ⊢ (∃𝑤∀𝑥(⊤ → 𝑥 = 𝑤) → 𝑦 = 𝑧) |
9 | 1, 8 | sylbi 219 | 1 ⊢ (∃*𝑥⊤ → 𝑦 = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ⊤wtru 1538 ∃wex 1780 ∃*wmo 2620 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-mo 2622 |
This theorem is referenced by: wl-motae 34770 |
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