Theorem 19.22d 744

Description: Deduction from Theorem 19.22 of [Margaris] p. 90.

Hypotheses

Ref	Expression
19.22d.1	⊢ (φ → ∀xφ)
19.22d.2	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
19.22d	⊢ (φ → (∃xψ → ∃xχ))

Proof of Theorem 19.22d

Step	Hyp	Ref	Expression
1		19.22d.1	. . 3 ⊢ (φ → ∀xφ)
2		19.22d.2	. . 3 ⊢ (φ → (ψ → χ))
3	1, 2	19.21ai 740	. 2 ⊢ (φ → ∀x(ψ → χ))
4		19.22 722	. 2 ⊢ (∀x(ψ → χ) → (∃xψ → ∃xχ))
5	3, 4	syl 12	1 ⊢ (φ → (∃xψ → ∃xχ))

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

This theorem is referenced by:  hbexd 791  exintr 793  19.22dv 947  mopick2 1057  ssopab2 2119  dmcosseq 2572  suppsr2 4017

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-gen 677

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

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