Theorem sbievw 2128

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2534 and sbiev 2347 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Aug-2025.)

Hypothesis

Ref	Expression
sbievw.is	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

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

Proof of Theorem sbievw

Step	Hyp	Ref	Expression
1		sbievw.is	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	sbbiiev 2127	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)
3		sbv 2122	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
4	2, 3	bitri 278	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  [wsb 2096

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097

This theorem is used by:  sbiedvw  2130  2sbievw  2131  sbievw2  2133  cbvsbv  2135  sbco4  2137  sbid2vw  2295  eqabbw  2836  sbralie  3342  sbralieALT  3343  rabrabi  3435  elabgw  3636  ralab  3656  sbcco2  3771  sbcie2g  3784  csbied  3889  dfss2  3923  unabw  4260  notabw  4266  2reu8i  47878  ichcircshi  48231