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Theorem sb7f 2452
Description: This version of dfsb7 2454 does not require that 𝜑 and 𝑧 be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1836 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1878 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
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 2385 . . 3 ([𝑧 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑧𝜑))
32sbbii 1884 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧𝜑))
41sbco2 2414 . 2 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
5 sb5 2429 . 2 ([𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧𝜑) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
63, 4, 53bitr3i 290 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1701  wnf 1705  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  sb7h  2453  dfsb7  2454
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