Theorem r19.23ad 1285

Description: Deduction from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.)

Hypotheses

Ref	Expression
r19.23ad.1	⊢ (φ → ∀xφ)
r19.23ad.2	⊢ (χ → ∀xχ)
r19.23ad.3	⊢ (φ → (x ∈ A → (ψ → χ)))

Assertion

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

Proof of Theorem r19.23ad

Step	Hyp	Ref	Expression
1		r19.23ad.1	. . 3 ⊢ (φ → ∀xφ)
2		r19.23ad.2	. . 3 ⊢ (χ → ∀xχ)
3		r19.23ad.3	. . . 4 ⊢ (φ → (x ∈ A → (ψ → χ)))
4	3	imp3a 279	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) → χ))
5	1, 2, 4	19.23ad 748	. 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) → χ))
6		df-rex 1206	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
7	5, 6	syl5ib 181	1 ⊢ (φ → (∃x ∈ A ψ → χ))

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

This theorem is referenced by:  reuuni4 1959  onfr 2237  peano5 2394  ffnfv 2892  iunon 2947  iinon 2948  tz7.49 2997  nneneq 3408  zornlem4 3606  zornlem5 3607  atom1d 5750

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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