Theorem sbiev 2347

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2533 with a disjoint variable condition, not requiring ax-13 2403. 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 2306	. 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 2215

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  2348  sbco2v  2363  mo4f  2594  reu2  3686  rmo4f  3696  sbcralt  3822  sbcreu  3826  sbcel12  4372  sbceqg  4373  sbcbr123  5163  frpoins2fg  6346  tfis2f  7856  tfinds  7860  setinds2f  9733  frins2f  9739  scottabf  9882  clwwlknonclwlknonf1o  30850  dlwwlknondlwlknonf1o  30853  funcnv4mpt  33149  nn0min  33299  ballotlemodife  35017  bnj1321  35544  bj-sbeqALT  37651