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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 1615 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 1445 | . . . . 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 1359 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓)) |
7 | 2 | a1d 22 | . . . 4 ⊢ (𝜓 → (𝑥 = 𝑦 → ∀𝑥𝜓)) |
8 | 2, 7 | alrimih 1358 | . . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓)) |
9 | ax9o 1588 | . . 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 1241 |
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 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: sb6x 1662 dvelimfALT2 1698 dvelimALT 1886 dvelimfv 1887 dvelimor 1894 |
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