Theorem 19.23ad 748

Description: Deduction from Theorem 19.23 of [Margaris] p. 90.

Hypotheses

Ref	Expression
19.23ad.1	⊢ (φ → ∀xφ)
19.23ad.2	⊢ (χ → ∀xχ)
19.23ad.3	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
19.23ad	⊢ (φ → (∃xψ → χ))

Proof of Theorem 19.23ad

Step	Hyp	Ref	Expression
1		19.23ad.1	. . 3 ⊢ (φ → ∀xφ)
2		19.23ad.3	. . 3 ⊢ (φ → (ψ → χ))
3	1, 2	19.21ai 740	. 2 ⊢ (φ → ∀x(ψ → χ))
4		19.23ad.2	. . 3 ⊢ (χ → ∀xχ)
5	4	19.23 745	. 2 ⊢ (∀x(ψ → χ) ↔ (∃xψ → χ))
6	3, 5	sylib 173	1 ⊢ (φ → (∃xψ → χ))

Syntax hints:   → wi 2  ∀wal 672  ∃wex 678

This theorem is referenced by:  19.23adv 954  r19.23ad 1285  alexeq 1409  dffun7 2688  fopab2 2891  cbvfo 2923  tz7.48-1 2994

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

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