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| Description: This version of dfsb7 2278 does not require that 𝜑 and 𝑧 be disjoint. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1909, i.e., that does not have the concept of a variable not occurring in a formula. (Definition dfsb1 2485 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| sb7f.1 | ⊢ Ⅎ𝑧𝜑 | 
| Ref | Expression | 
|---|---|
| sb7f | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sb7f.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
| 2 | 1 | sb5f 2502 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) | 
| 3 | 2 | sbbii 2075 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) | 
| 4 | 1 | sbco2 2515 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | 
| 5 | sb5 2275 | . 2 ⊢ ([𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | |
| 6 | 3, 4, 5 | 3bitr3i 301 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1778 Ⅎwnf 1782 [wsb 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 | 
| This theorem is referenced by: sb7h 2530 | 
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