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Metamath Proof Explorer

Theorem sb7f

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) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypothesis sb7f.1 ⊢ Ⅎ 𝑧 𝜑
Assertion sb7f ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑧 ( 𝑧 = 𝑦 ∧ ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 sb7f.1 ⊢ Ⅎ 𝑧 𝜑
2 1 sb5f ⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝜑 ) )
3 2 sbbii ⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑧 ] ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝜑 ) )
4 1 sbco2 ⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
5 sb5 ⊢ ( [ 𝑦 / 𝑧 ] ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝜑 ) ↔ ∃ 𝑧 ( 𝑧 = 𝑦 ∧ ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝜑 ) ) )
6 3 4 5 3bitr3i ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑧 ( 𝑧 = 𝑦 ∧ ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝜑 ) ) )