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Mirrors > Home > MPE Home > Th. List > albi | Structured version Visualization version GIF version |
Description: Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.) |
Ref | Expression |
---|---|
albi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp 217 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | al2imi 1816 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
3 | biimpr 222 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
4 | 3 | al2imi 1816 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜓 → ∀𝑥𝜑)) |
5 | 2, 4 | impbid 214 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 |
This theorem is referenced by: albii 1820 nfbiit 1851 albidh 1867 19.16 2227 19.17 2228 equvel 2479 eqeq1d 2823 intmin4 4905 dfiin2g 4957 eunex 5291 bj-2albi 33947 bj-hbxfrbi 33963 bj-sblem 34168 wl-aleq 34790 2albi 40730 ralbidar 40797 trsbcVD 41231 sbcssgVD 41237 ichal 43647 |
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