Theorem sbhypf 3539

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3923. (Contributed by Raph Levien, 10-Apr-2004.) (Proof shortened by Wolf Lammen, 25-Jan-2025.)

Hypotheses

Ref	Expression
sbhypf.1	⊢ Ⅎ𝑥𝜓
sbhypf.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbhypf	⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦

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

Proof of Theorem sbhypf

Step	Hyp	Ref	Expression
1		sbhypf.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	sbimi 2078	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥](𝜑 ↔ 𝜓))
3		eqsb1 2860	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
4		sbhypf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	sbf 2263	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
6	5	sblbis 2306	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
7	2, 3, 6	3imtr3i 291	1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542  Ⅎwnf 1786  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-cleq 2725

This theorem is referenced by:  mob2  3712  reu2eqd  3733  cbvrabcsfw  3938  cbvopab1  5224  ralxpf  5847  cbviotaw  6503  cbvriotaw  7374  tfisi  7848  ac6sf  10484  nn0ind-raph  12662  ac6sf2  31849  nn0min  32026  ac6gf  36600  fdc1  36614