Theorem rspsbc 3901

Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 2068 and spsbc 3817. See also rspsbca 3902 and rspcsbela 4461. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
rspsbc	⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rspsbc

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsvw 3323	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑)
2		dfsbcq2 3807	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	rspcv 3631	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
4	1, 3	biimtrid 242	1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4  [wsb 2064   ∈ wcel 2108  ∀wral 3067  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-sbc 3805

This theorem is referenced by:  rspsbca  3902  sbcth2  3906  rspcsbela  4461  riota5f  7433  riotass2  7435  fzrevral  13669  fprodcllemf  16006  rspcsbnea  42088  rspsbc2  44505  truniALT  44512  rspsbc2VD  44826  truniALTVD  44849  trintALTVD  44851  trintALT  44852