Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > albidh | Structured version Visualization version GIF version |
Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.) |
Ref | Expression |
---|---|
albidh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
albidh.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
albidh | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albidh.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | albidh.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | alrimih 1741 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | albi 1736 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 |
This theorem is referenced by: albidv 1836 albid 2077 albidOLD 2187 dral2-o 33233 ax12indalem 33248 ax12inda2ALT 33249 ax12inda 33251 |
Copyright terms: Public domain | W3C validator |