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Mirrors > Home > MPE Home > Th. List > Mathboxes > equidq | Structured version Visualization version GIF version |
Description: equid 2024 with universal quantifier without using ax-c5 36540 or ax-5 1917. (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 36579 | . 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 | |
2 | ax10fromc7 36552 | . . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥) | |
3 | hbequid 36566 | . . . 4 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) | |
4 | 3 | con3i 157 | . . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥) |
5 | 2, 4 | alrimih 1830 | . 2 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥) |
6 | 1, 5 | mt3 204 | 1 ⊢ ∀𝑦 𝑥 = 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-c5 36540 ax-c4 36541 ax-c7 36542 ax-c10 36543 ax-c9 36547 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 |
This theorem is referenced by: (None) |
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