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| Mirrors > Home > MPE Home > Th. List > axc11 | Structured version Visualization version GIF version | ||
| Description: Show that ax-c11 38888 can be derived from ax-c11n 38889 in the form of axc11n 2431. Normally, axc11 2435 should be used rather than ax-c11 38888, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker axc11v 2264 when possible. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| axc11 | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | axc11r 2371 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) | |
| 2 | 1 | aecoms 2433 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: hbae 2436 dral1 2444 dral1ALT 2445 nd1 10627 nd2 10628 axc11n11 36683 bj-hbaeb2 36819 wl-aetr 37530 ax6e2eq 44577 ax6e2eqVD 44927 2sb5ndVD 44930 2sb5ndALT 44952 |
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