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Mirrors > Home > ILE Home > Th. List > sb5f | GIF version |
Description: Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) |
Ref | Expression |
---|---|
equs45f.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
sb5f | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equs45f.1 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | sb6f 1776 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
3 | 1 | equs45f 1775 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
4 | 2, 3 | bitr4i 186 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1330 ∃wex 1469 [wsb 1736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-11 1485 ax-4 1488 ax-i9 1511 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-sb 1737 |
This theorem is referenced by: sbcof2 1783 |
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