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Metamath Proof Explorer

Theorem cbvmptg

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 . See cbvmpt for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by NM, 11-Sep-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvmptg.1	⊢ Ⅎ 𝑦 𝐵
	cbvmptg.2	⊢ Ⅎ 𝑥 𝐶
	cbvmptg.3	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
Assertion	cbvmptg	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		cbvmptg.1	⊢ Ⅎ 𝑦 𝐵
2		cbvmptg.2	⊢ Ⅎ 𝑥 𝐶
3		cbvmptg.3	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
4		nfcv	⊢ Ⅎ 𝑥 𝐴
5		nfcv	⊢ Ⅎ 𝑦 𝐴
6	4 5 1 2 3	cbvmptfg	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ 𝐶 )