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Mirrors > Home > MPE Home > Th. List > sb5rf | Structured version Visualization version GIF version |
Description: Reversed substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb5rf.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb5rf | ⊢ (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5rf.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | sbequ12r 2249 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 1, 2 | equsex 2420 | . 2 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) ↔ 𝜑) |
4 | 3 | bicomi 223 | 1 ⊢ (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∃wex 1786 Ⅎwnf 1790 [wsb 2071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-12 2175 ax-13 2374 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-nf 1791 df-sb 2072 |
This theorem is referenced by: (None) |
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