Theorem 19.9d 720

Description: A deduction version of one direction of 19.9r 718.

Hypotheses

Ref	Expression
19.9d.1	⊢ (ψ → ∀xψ)
19.9d.2	⊢ (ψ → (φ → ∀xφ))

Assertion

Ref	Expression
19.9d	⊢ (ψ → (∃xφ → φ))

Proof of Theorem 19.9d

Step	Hyp	Ref	Expression
1		19.9d.1	. 2 ⊢ (ψ → ∀xψ)
2		19.9d.2	. . 3 ⊢ (ψ → (φ → ∀xφ))
3	2	19.20i 691	. 2 ⊢ (∀xψ → ∀x(φ → ∀xφ))
4		19.9t 719	. 2 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))
5	1, 3, 4	3syl 21	1 ⊢ (ψ → (∃xφ → φ))

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

This theorem is referenced by:  sbequi 876

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-ex 679

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