Theorem sb7h

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

	Ref	Expression
Hypothesis	sb7h.1	⊢ ( 𝜑 → ∀ 𝑧 𝜑 )
Assertion	sb7h	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑧 ( 𝑧 = 𝑦 ∧ ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝜑 ) ) )

Proof

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