Theorem r19.23ai 1283

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.)

Hypotheses

Ref	Expression
r19.23ai.1	⊢ (ψ → ∀xψ)
r19.23ai.2	⊢ (x ∈ A → (φ → ψ))

Assertion

Ref	Expression
r19.23ai	⊢ (∃x ∈ A φ → ψ)

Proof of Theorem r19.23ai

Step	Hyp	Ref	Expression
1		df-rex 1206	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
2		r19.23ai.1	. . 3 ⊢ (ψ → ∀xψ)
3		r19.23ai.2	. . . 4 ⊢ (x ∈ A → (φ → ψ))
4	3	imp 277	. . 3 ⊢ ((x ∈ A ∧ φ) → ψ)
5	2, 4	19.23ai 746	. 2 ⊢ (∃x(x ∈ A ∧ φ) → ψ)
6	1, 5	sylbi 174	1 ⊢ (∃x ∈ A φ → ψ)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  r19.23aiv 1284  tfinds 2401  r1val1 3502  rankuni 3533

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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