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Mirrors > Home > MPE Home > Th. List > equs5 | Structured version Visualization version GIF version |
Description: Lemma used in proofs of substitution properties. If there is a disjoint variable condition on 𝑥, 𝑦, then sb56 2307 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2479 can be used. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.) |
Ref | Expression |
---|---|
equs5 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfna1 2205 | . . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | nfa1 2204 | . . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
3 | axc15 2442 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | |
4 | 3 | impd 400 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
5 | 1, 2, 4 | exlimd 2263 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
6 | equs4 2436 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
7 | 5, 6 | impbid1 217 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∀wal 1656 ∃wex 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-10 2194 ax-12 2222 ax-13 2390 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ex 1881 df-nf 1885 |
This theorem is referenced by: sb3 2484 sb4 2486 bj-sbsb 33347 |
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