Theorem rspsbc 3832

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 2097 and spsbc 3757. See also rspsbca 3833 and rspcsbela 4391. (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 3312	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑)
2		dfsbcq2 3747	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	rspcv 3577	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
4	1, 3	biimtrid 244	1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4  [wsb 2089   ∈ wcel 2141  ∀wral 3075  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-sbc 3745

This theorem is referenced by:  rspsbca  3833  sbcth2  3837  rspcsbela  4391  riota5f  7377  riotass2  7379  fzrevral  13614  fprodcllemf  15971  rspcsbnea  42712  rspsbc2  45074  truniALT  45081  rspsbc2VD  45394  truniALTVD  45417  trintALTVD  45419  trintALT  45420