Theorem sbiev 2319

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2506 with a disjoint variable condition, not requiring ax-13 2376. See sbievw 2099 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 2147 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 2098	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)
3		sbiev.1	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	sbf 2278	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
5	2, 4	bitri 275	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  Ⅎwnf 1785  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-sb 2069

This theorem is referenced by:  sbiedw  2321  sbco2v  2336  mo4f  2567  cbvrabwOLD  3425  reu2  3671  rmo4f  3681  sbcralt  3810  sbcreu  3814  sbcel12  4351  sbceqg  4352  sbcbr123  5139  frpoins2fg  6308  tfis2f  7807  tfinds  7811  setinds2f  9671  frins2f  9677  clwwlknonclwlknonf1o  30432  dlwwlknondlwlknonf1o  30435  funcnv4mpt  32741  nn0min  32894  ballotlemodife  34642  bnj1321  35169  bj-sbeqALT  37207  scottabf  44667