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Mirrors > Home > MPE Home > Th. List > axc11r | Structured version Visualization version GIF version |
Description: Same as axc11 2456 but with reversed antecedent. Note the use of ax-12 2196 (and not merely ax12v 2197). (Contributed by NM, 25-Jul-2015.) |
Ref | Expression |
---|---|
axc11r | ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-12 2196 | . . 3 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) | |
2 | 1 | sps 2202 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) |
3 | pm2.27 42 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑)) | |
4 | 3 | al2imi 1892 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑)) |
5 | 2, 4 | syld 47 | 1 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-12 2196 |
This theorem depends on definitions: df-bi 197 df-ex 1854 |
This theorem is referenced by: ax12 2449 axc11n 2451 axc11nOLD 2452 axc11 2456 hbae 2457 dral1 2465 dral1ALT 2466 axpowndlem3 9613 axc11n11r 32979 bj-ax12v3ALT 32982 bj-axc11v 33053 bj-dral1v 33054 bj-hbaeb2 33111 |
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