Theorem cbvsbv 2138

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 2835 version. (Contributed by Wolf Lammen, 16-Mar-2025.)

Hypothesis

Ref	Expression
cbvsbv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvsbv	⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥, 𝑧)   𝜓(𝑦, 𝑧)

Proof of Theorem cbvsbv

Step	Hyp	Ref	Expression
1		sbco2vv 2137	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
2		cbvsbv.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	sbievw 2131	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
4	3	sbbii 2113	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
5	1, 4	bitr3i 280	1 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  sbco4lem  2139  cbvabv  2835