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Mirrors > Home > ILE Home > Th. List > equsalh | GIF version |
Description: A useful equivalence related to substitution. New proofs should use equsal 1612 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equsalh.1 | ⊢ (ψ → ∀xψ) |
equsalh.2 | ⊢ (x = y → (φ ↔ ψ)) |
Ref | Expression |
---|---|
equsalh | ⊢ (∀x(x = y → φ) ↔ ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalh.2 | . . . . 5 ⊢ (x = y → (φ ↔ ψ)) | |
2 | equsalh.1 | . . . . . 6 ⊢ (ψ → ∀xψ) | |
3 | 2 | 19.3h 1442 | . . . . 5 ⊢ (∀xψ ↔ ψ) |
4 | 1, 3 | syl6bbr 187 | . . . 4 ⊢ (x = y → (φ ↔ ∀xψ)) |
5 | 4 | pm5.74i 169 | . . 3 ⊢ ((x = y → φ) ↔ (x = y → ∀xψ)) |
6 | 5 | albii 1356 | . 2 ⊢ (∀x(x = y → φ) ↔ ∀x(x = y → ∀xψ)) |
7 | 2 | a1d 22 | . . . 4 ⊢ (ψ → (x = y → ∀xψ)) |
8 | 2, 7 | alrimih 1355 | . . 3 ⊢ (ψ → ∀x(x = y → ∀xψ)) |
9 | ax9o 1585 | . . 3 ⊢ (∀x(x = y → ∀xψ) → ψ) | |
10 | 8, 9 | impbii 117 | . 2 ⊢ (ψ ↔ ∀x(x = y → ∀xψ)) |
11 | 6, 10 | bitr4i 176 | 1 ⊢ (∀x(x = y → φ) ↔ ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: sb6x 1659 dvelimfALT2 1695 dvelimALT 1883 dvelimfv 1884 dvelimor 1891 |
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