Theorem sbiev 2345

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2531 with a disjoint variable condition, not requiring ax-13 2401. See sbievw 2131 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2178 and shorten proof. (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Jul-2025.)

Hypotheses

Ref	Expression
sbiev.1	⊢ Ⅎ𝑥𝜓
sbiev.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbiev	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbiev

Step	Hyp	Ref	Expression
1		sbiev.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	sbbiiev 2130	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)
3		sbiev.1	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	sbf 2304	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
5	2, 4	bitri 278	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  Ⅎwnf 1816  [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  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  sbiedw  2346  sbco2v  2361  mo4f  2592  reu2  3683  rmo4f  3693  sbcralt  3819  sbcreu  3823  sbcel12  4369  sbceqg  4370  sbcbr123  5159  frpoins2fg  6342  tfis2f  7852  tfinds  7856  setinds2f  9729  frins2f  9735  scottabf  9878  clwwlknonclwlknonf1o  30842  dlwwlknondlwlknonf1o  30845  funcnv4mpt  33141  nn0min  33291  ballotlemodife  35009  bnj1321  35536  bj-sbeqALT  37643