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Metamath Proof Explorer

Theorem dfsbcq2

Description: This theorem, which is similar to Theorem 6.7 of Quine p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb and substitution for class variables df-sbc . Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq . (Contributed by NM, 31-Dec-2016)

		Ref	Expression
	Assertion	dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝐴 ∈ { 𝑥 ∣ 𝜑 } ) )
2		df-clab	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝑦 / 𝑥 ] 𝜑 )
3		df-sbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜑 } )
4	3	bicomi	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝐴 / 𝑥 ] 𝜑 )
5	1 2 4	3bitr3g	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )