Theorem ra4sbca 1994

Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 69.

Assertion

Ref	Expression
ra4sbca	⊢ ((A ∈ B ⋀ ∀x ∈ B φ) → [A / x]φ)

Distinct variable group:   x,B

Proof of Theorem ra4sbca

Step	Hyp	Ref	Expression
1		ra4sbc 1993	. 2 ⊢ (A ∈ B → (∀x ∈ B φ → [A / x]φ))
2	1	imp 350	1 ⊢ ((A ∈ B ⋀ ∀x ∈ B φ) → [A / x]φ)

Syntax hints:   → wi 3   ⋀ wa 223   ∈ wcel 956  [wsbc 1168  ∀wral 1642

This theorem is referenced by:  fsump1s 6959  fsumcmp 6986

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-sbc 1938

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