Theorem sb2ae 2499

Description: In the case of two successive substitutions for two always equal variables, the second substitution has no effect. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by BJ and WL, 9-Aug-2023.) (New usage is discouraged.)

Assertion

Ref	Expression
sb2ae	⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))

Distinct variable group:   𝑦,𝑣

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem sb2ae

Step	Hyp	Ref	Expression
1		drsb1 2498	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑦][𝑣 / 𝑦]𝜑))
2		nfs1v 2157	. . 3 ⊢ Ⅎ𝑦[𝑣 / 𝑦]𝜑
3	2	sbf 2267	. 2 ⊢ ([𝑢 / 𝑦][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
4	1, 3	bitrdi 290	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-sb 2071

This theorem is referenced by: (None)