Theorem sbiev 2346

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2532 with a disjoint variable condition, not requiring ax-13 2402. 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 2305	. 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  2347  sbco2v  2362  mo4f  2593  reu2  3683  rmo4f  3693  sbcralt  3819  sbcreu  3823  sbcel12  4369  sbceqg  4370  sbcbr123  5159  frpoins2fg  6340  tfis2f  7856  tfinds  7860  setinds2f  9735  frins2f  9741  scottabf  9920  clwwlknonclwlknonf1o  30945  dlwwlknondlwlknonf1o  30948  funcnv4mpt  33244  nn0min  33394  ballotlemodife  35113  bnj1321  35640  bj-sbeqALT  37782