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Mirrors > Home > MPE Home > Th. List > equsalh | Structured version Visualization version GIF version |
Description: An equivalence related to implicit substitution. See equsalhw 2291 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 2-Jun-1993.) |
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.1 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | 1 | nf5i 2141 | . 2 ⊢ Ⅎ𝑥𝜓 |
3 | equsalh.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | equsal 2433 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 ax-13 2383 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 |
This theorem is referenced by: dvelimf-o 35947 |
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