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Theorem sb7f 2546
 Description: This version of dfsb7 2282 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 1912, i.e., that does not have the concept of a variable not occurring in a formula. (Definition dfsb1 2500 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 2380. (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
sb7f.1 𝑧𝜑
Assertion
Ref Expression
sb7f ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sb7f
StepHypRef Expression
1 sb7f.1 . . . 4 𝑧𝜑
21sb5f 2517 . . 3 ([𝑧 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑧𝜑))
32sbbii 2082 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧𝜑))
41sbco2 2531 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
5 sb5 2274 . 2 ([𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
63, 4, 53bitr3i 304 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1782  Ⅎwnf 1786  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by:  sb7h  2547
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