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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 205 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | al2imi 1743 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
3 | biimpr 210 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
4 | 3 | al2imi 1743 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜓 → ∀𝑥𝜑)) |
5 | 2, 4 | impbid 202 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 |
This theorem depends on definitions: df-bi 197 |
This theorem is referenced by: albii 1747 nfbiit 1777 albidh 1793 19.16 2093 19.17 2094 equvel 2347 eqeq1d 2624 intmin4 4506 dfiin2g 4553 bj-2albi 32597 bj-hbxfrbi 32608 wl-aleq 33322 2albi 38577 ralbidar 38649 sbcssOLD 38756 trsbcVD 39113 sbcssgVD 39119 |
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