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Mirrors > Home > MPE Home > Th. List > equvini | Structured version Visualization version GIF version |
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦. See equvinv 1946 for a shorter proof requiring fewer axioms when 𝑧 is required to be distinct from 𝑥 and 𝑦. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) |
Ref | Expression |
---|---|
equvini | ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtr 1935 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → 𝑧 = 𝑦)) | |
2 | equeuclr 1937 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧)) | |
3 | 2 | anc2ri 579 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦))) |
4 | 1, 3 | syli 38 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦))) |
5 | 19.8a 2039 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) | |
6 | 4, 5 | syl6 34 | . 2 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))) |
7 | ax13 2237 | . . 3 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) | |
8 | ax6e 2238 | . . . . 5 ⊢ ∃𝑧 𝑧 = 𝑦 | |
9 | 8, 3 | eximii 1754 | . . . 4 ⊢ ∃𝑧(𝑥 = 𝑦 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
10 | 9 | 19.35i 1795 | . . 3 ⊢ (∀𝑧 𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
11 | 7, 10 | syl6 34 | . 2 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))) |
12 | 6, 11 | pm2.61i 175 | 1 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
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-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: 2ax6elem 2437 |
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