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Metamath Proof Explorer

Theorem pm13.195

Description: Theorem *13.195 in WhiteheadRussell p. 179. This theorem is very similar to sbc5 . (Contributed by Andrew Salmon, 3-Jun-2011) (Revised by NM, 4-Jan-2017)

		Ref	Expression
	Assertion	pm13.195	⊢ ( ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 ) ↔ [ 𝐴 / 𝑦 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbc5	⊢ ( [ 𝐴 / 𝑦 ] 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 ) )
2	1	bicomi	⊢ ( ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 ) ↔ [ 𝐴 / 𝑦 ] 𝜑 )