Theorem cbvexd 978

Description: Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with ddelim 1000.

Hypotheses

Ref	Expression
cbvald.1	⊢ (φ → ∀yφ)
cbvald.2	⊢ (φ → (ψ → ∀yψ))
cbvald.3	⊢ (φ → (x = y → (ψ ↔ χ)))

Assertion

Ref	Expression
cbvexd	⊢ (φ → (∃xψ ↔ ∃yχ))

Distinct variable group(s):   φ,x   χ,x

Proof of Theorem cbvexd

Step	Hyp	Ref	Expression
1		cbvald.1	. . . 4 ⊢ (φ → ∀yφ)
2		cbvald.2	. . . . 5 ⊢ (φ → (ψ → ∀yψ))
3	1, 2	hbnd 786	. . . 4 ⊢ (φ → (¬ ψ → ∀y ¬ ψ))
4		cbvald.3	. . . . 5 ⊢ (φ → (x = y → (ψ ↔ χ)))
5		pm4.11 400	. . . . 5 ⊢ ((ψ ↔ χ) ↔ (¬ ψ ↔ ¬ χ))
6	4, 5	syl6ib 185	. . . 4 ⊢ (φ → (x = y → (¬ ψ ↔ ¬ χ)))
7	1, 3, 6	cbvald 977	. . 3 ⊢ (φ → (∀x ¬ ψ ↔ ∀y ¬ χ))
8	7	negbid 463	. 2 ⊢ (φ → (¬ ∀x ¬ ψ ↔ ¬ ∀y ¬ χ))
9		df-ex 679	. 2 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
10		df-ex 679	. 2 ⊢ (∃yχ ↔ ¬ ∀y ¬ χ)
11	8, 9, 10	3bitr4g 428	1 ⊢ (φ → (∃xψ ↔ ∃yχ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = weq 797

This theorem is referenced by:  axrepndlem2 3739  axunnd 3742  axpowndlem2 3744  axpownd 3747  axregndlem2 3749  axinfndlem1 3751  axacndlem4 3756

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925

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

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