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Metamath Proof Explorer

Theorem cbvmptvg

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbvmptv for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 19-Feb-2013) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	cbvmptvg.1	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
	Assertion	cbvmptvg	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ 𝐶 )

Proof

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