Theorem sbhypf 3502

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3877. (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 2079	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 → [𝑦 / 𝑥](𝜑 ↔ 𝜓))
3		eqsb1 2862	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
4		sbhypf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	sbf 2277	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
6	5	sblbis 2314	. 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 206   = wceq 1541  Ⅎwnf 1784  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-sb 2068  df-cleq 2728

This theorem is referenced by:  mob2  3673  reu2eqd  3694  cbvrabcsfw  3890  cbvopab1  5172  ralxpf  5795  cbviotaw  6455  cbvriotaw  7324  tfisi  7801  ac6sf  10399  nn0ind-raph  12592  ac6sf2  32700  nn0min  32901  ac6gf  37933  fdc1  37947