Theorem sbievw 2093

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2507 and sbiev 2314 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 2092	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)
3		sbv 2088	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
4	2, 3	bitri 275	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 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:  sbiedvw  2095  2sbievw  2096  sbievw2  2098  cbvsbv  2100  sbco4  2102  sbid2vw  2259  2mosOLD  2650  eqabbw  2815  sbralieALT  3359  rabrabi  3456  elabgw  3677  ralab  3697  sbcco2  3815  sbcie2g  3829  csbied  3935  dfss2  3969  unabw  4307  notabw  4313  ab0w  4379  ab0orv  4383  2reu8i  47125  ichcircshi  47441