Theorem rspsbc 3125

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 2024 and spsbc 3059. See also rspsbca 3126 and rspcsbela 3196. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
rspsbc	|- (A e. B -> (A.x e. B ph -> [.A / x ].ph))

Distinct variable group:   x,B

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

Proof of Theorem rspsbc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 2847	. 2 |- (A.x e. B ph <-> A.y e. B [y / x]ph)
2		dfsbcq2 3050	. . 3 |- (y = A -> ([y / x]ph <-> [.A / x ].ph))
3	2	rspcv 2952	. 2 |- (A e. B -> (A.y e. B [y / x]ph -> [.A / x ].ph))
4	1, 3	syl5bi 208	1 |- (A e. B -> (A.x e. B ph -> [.A / x ].ph))

Syntax hints:   -> wi 4  [wsb 1648   e. wcel 1710  A.wral 2615   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048

This theorem is referenced by:  rspsbca  3126  sbcth2  3130  rspcsbela  3196