Theorem cbvsbv

Description: Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was extracted from a former cbvabv version. (Contributed by Wolf Lammen, 16-Mar-2025)

	Ref	Expression
Hypothesis	cbvsbv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvsbv	⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvsbv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		sbco2vv	⊢ ( [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 )
3	1	sbievw	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
4	3	sbbii	⊢ ( [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜓 )
5	2 4	bitr3i	⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜓 )