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Metamath Proof Explorer

Theorem sb5rf

Description: Reversed substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 3-Feb-2005) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 20-Sep-2018) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sb5rf.1	⊢ Ⅎ 𝑦 𝜑
	Assertion	sb5rf	⊢ ( 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝑥 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sb5rf.1	⊢ Ⅎ 𝑦 𝜑
2		sbequ12r	⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	1 2	equsex	⊢ ( ∃ 𝑦 ( 𝑦 = 𝑥 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ↔ 𝜑 )
4	3	bicomi	⊢ ( 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝑥 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) )