Theorem sbhypf 3514

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3881. (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 2108	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥](𝜑 ↔ 𝜓))
3		eqsb1 2889	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
4		sbhypf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	sbf 2306	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
6	5	sblbis 2343	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
7	2, 3, 6	3imtr3i 294	1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  Ⅎwnf 1813  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814  df-sb 2097  df-cleq 2755

This theorem is referenced by:  mob2  3678  reu2eqd  3699  cbvrabcsfw  3894  cbvopab1  5185  cbvmptf  5211  ralxpf  5832  cbviotaw  6499  cbvriotaw  7376  tfisi  7851  ac6sf  10468  nn0ind-raph  12691  ac6sf2  32967  nn0min  33165  ac6gf  38403  fdc1  38417