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Mirrors > Home > MPE Home > Th. List > albid | Structured version Visualization version GIF version |
Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
albid.1 | ⊢ Ⅎ𝑥𝜑 |
albid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
albid | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nf5ri 2053 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | albid.2 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | albidh 1780 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 |
This theorem is referenced by: nfbidf 2079 axc15 2291 dral2 2312 dral1 2313 ax12v2OLD 2330 sbal1 2448 sbal2 2449 eubid 2476 ralbida 2965 raleqf 3111 intab 4442 fin23lem32 9049 axrepndlem1 9293 axrepndlem2 9294 axrepnd 9295 axunnd 9297 axpowndlem2 9299 axpowndlem4 9301 axregndlem2 9304 axinfndlem1 9306 axinfnd 9307 axacndlem4 9311 axacndlem5 9312 axacnd 9313 funcnvmptOLD 28850 iota5f 30861 bj-dral1v 31936 wl-equsald 32504 wl-sbnf1 32515 wl-2sb6d 32520 wl-sbalnae 32524 wl-mo2df 32531 wl-eudf 32533 wl-ax11-lem6 32546 wl-ax11-lem8 32548 ax12eq 33244 ax12el 33245 ax12v2-o 33252 |
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