Theorem a4c1 844

Description: A more general version of a4c 843.

Hypotheses

Ref	Expression
a4c1.1	⊢ (χ → ∀xχ)
a4c1.2	⊢ (χ → (φ → ∀xφ))
a4c1.3	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
a4c1	⊢ (χ → (φ → ∃xψ))

Proof of Theorem a4c1

Step	Hyp	Ref	Expression
1		a4c1.1	. . . . . 6 ⊢ (χ → ∀xχ)
2	1	adantr 306	. . . . 5 ⊢ ((χ ∧ φ) → ∀xχ)
3		a4c1.2	. . . . . 6 ⊢ (χ → (φ → ∀xφ))
4	3	imp 277	. . . . 5 ⊢ ((χ ∧ φ) → ∀xφ)
5	2, 4	jca 236	. . . 4 ⊢ ((χ ∧ φ) → (∀xχ ∧ ∀xφ))
6		19.26 749	. . . 4 ⊢ (∀x(χ ∧ φ) ↔ (∀xχ ∧ ∀xφ))
7	5, 6	sylibr 175	. . 3 ⊢ ((χ ∧ φ) → ∀x(χ ∧ φ))
8		a4c1.3	. . . 4 ⊢ (x = y → (φ → ψ))
9	8	adantld 307	. . 3 ⊢ (x = y → ((χ ∧ φ) → ψ))
10	7, 9	a4c 843	. 2 ⊢ ((χ ∧ φ) → ∃xψ)
11	10	exp 291	1 ⊢ (χ → (φ → ∃xψ))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797

This theorem is referenced by:  eqvin.l1 851

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  ax-9 799

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

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