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Mirrors > Home > NFE Home > Th. List > dral1 | GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.) |
Ref | Expression |
---|---|
dral1.1 | ⊢ (∀x x = y → (φ ↔ ψ)) |
Ref | Expression |
---|---|
dral1 | ⊢ (∀x x = y → (∀xφ ↔ ∀yψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbae 1953 | . . . 4 ⊢ (∀x x = y → ∀x∀x x = y) | |
2 | dral1.1 | . . . . 5 ⊢ (∀x x = y → (φ ↔ ψ)) | |
3 | 2 | biimpd 198 | . . . 4 ⊢ (∀x x = y → (φ → ψ)) |
4 | 1, 3 | alimdh 1563 | . . 3 ⊢ (∀x x = y → (∀xφ → ∀xψ)) |
5 | ax10o 1952 | . . 3 ⊢ (∀x x = y → (∀xψ → ∀yψ)) | |
6 | 4, 5 | syld 40 | . 2 ⊢ (∀x x = y → (∀xφ → ∀yψ)) |
7 | hbae 1953 | . . . 4 ⊢ (∀x x = y → ∀y∀x x = y) | |
8 | 2 | biimprd 214 | . . . 4 ⊢ (∀x x = y → (ψ → φ)) |
9 | 7, 8 | alimdh 1563 | . . 3 ⊢ (∀x x = y → (∀yψ → ∀yφ)) |
10 | ax10o 1952 | . . . 4 ⊢ (∀y y = x → (∀yφ → ∀xφ)) | |
11 | 10 | aecoms 1947 | . . 3 ⊢ (∀x x = y → (∀yφ → ∀xφ)) |
12 | 9, 11 | syld 40 | . 2 ⊢ (∀x x = y → (∀yψ → ∀xφ)) |
13 | 6, 12 | impbid 183 | 1 ⊢ (∀x x = y → (∀xφ ↔ ∀yψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 |
This theorem is referenced by: drex1 1967 drnf1 1969 equveli 1988 a16gALT 2049 sb9i 2094 ralcom2 2775 |
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