Theorem sb7 991

Description: An alternate definition of proper substitution df-sb 853. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and φ of the definiendum. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 1093.

Assertion

Ref	Expression
sb7	⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ)))

Distinct variable group(s):   x,z   y,z   φ,z

Proof of Theorem sb7

Step	Hyp	Ref	Expression
1		sb5 988	. . 3 ⊢ ([z / x]φ ↔ ∃x(x = z ∧ φ))
2	1	bisb 855	. 2 ⊢ ([y / z][z / x]φ ↔ [y / z]∃x(x = z ∧ φ))
3		ax-17 925	. . 3 ⊢ (φ → ∀zφ)
4	3	sbco2 913	. 2 ⊢ ([y / z][z / x]φ ↔ [y / x]φ)
5		sb5 988	. 2 ⊢ ([y / z]∃x(x = z ∧ φ) ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ)))
6	2, 4, 5	3bitr3 156	1 ⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ)))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797  [wsb 852

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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