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| Mirrors > Home > MPE Home > Th. List > Mathboxes > equidq | Structured version Visualization version GIF version | ||
| Description: equid 2010 with universal quantifier without using ax-c5 38885 or ax-5 1909. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| equidq | ⊢ ∀𝑦 𝑥 = 𝑥 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | equidqe 38924 | . 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 | |
| 2 | ax10fromc7 38897 | . . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥) | |
| 3 | hbequid 38911 | . . . 4 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) | |
| 4 | 3 | con3i 154 | . . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥) | 
| 5 | 2, 4 | alrimih 1823 | . 2 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥) | 
| 6 | 1, 5 | mt3 201 | 1 ⊢ ∀𝑦 𝑥 = 𝑥 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∀wal 1537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-c5 38885 ax-c4 38886 ax-c7 38887 ax-c10 38888 ax-c9 38892 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: (None) | 
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