Theorem rspcdf 3602

Description: Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.)

Hypotheses

Ref	Expression
rspcdf.1	⊢ Ⅎ𝑥𝜑
rspcdf.2	⊢ Ⅎ𝑥𝜒
rspcdf.3	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcdf.4	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rspcdf	⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rspcdf

Step	Hyp	Ref	Expression
1		rspcdf.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rspcdf.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3	2	ex 415	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
4	1, 3	alrimi 2213	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
5		rspcdf.3	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
6		rspcdf.2	. . 3 ⊢ Ⅎ𝑥𝜒
7	6	rspct 3601	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)))
8	4, 5, 7	sylc 65	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∀wral 3138

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3488

This theorem is referenced by:  rspc2daf  30215