Theorem sb7f 2521

Description: This version of dfsb7 2251 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 1888 i.e. that doesn't have the concept of a variable not occurring in a wff. (dfsb1 2464 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	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
sb7f	⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))

Distinct variable group:   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sb7f

Step	Hyp	Ref	Expression
1		sb7f.1	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	sb5f 2492	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))
3	2	sbbii 2054	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧 ∧ 𝜑))
4	1	sbco2 2507	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
5		sb5 2240	. 2 ⊢ ([𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))
6	3, 4, 5	3bitr3i 302	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∃wex 1761  Ⅎwnf 1765  [wsb 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043

This theorem is referenced by:  sb7h  2522  dfsb7OLDOLD  2523