Theorem sb7f 2556

Description: This version of dfsb7 2313 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 1930, i.e., that does not have the concept of a variable not occurring in a formula. (Definition dfsb1 2512 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 2403. (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)

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 2529	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))
3	2	sbbii 2109	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧 ∧ 𝜑))
4	1	sbco2 2542	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
5		sb5 2310	. 2 ⊢ ([𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))
6	3, 4, 5	3bitr3i 303	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1799  Ⅎwnf 1803  [wsb 2090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091

This theorem is referenced by:  sb7h  2557