Theorem sb7h 2558

Description: This version of dfsb7 2314 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 1940, i.e., that does not have the concept of a variable not occurring in a formula. (Definition dfsb1 2513 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 2404. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb7h.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
sb7h	⊢ ([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 sb7h

Step	Hyp	Ref	Expression
1		sb7h.1	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	nf5i 2181	. 2 ⊢ Ⅎ𝑧𝜑
3	2	sb7f 2557	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  ∃wex 1809  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097

This theorem is referenced by: (None)