Theorem cbvsbv 2100

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 2812 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 2099	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
2		cbvsbv.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	sbievw 2093	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
4	3	sbbii 2076	. 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
5	1, 4	bitr3i 277	1 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)

Syntax hints:   → wi 4   ↔ wb 206  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065

This theorem is referenced by:  sbco4lem  2101  cbvabv  2812