Theorem cbvexd 2009

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
cbvald.1	⊢ Ⅎyφ
cbvald.2	⊢ (φ → Ⅎyψ)
cbvald.3	⊢ (φ → (x = y → (ψ ↔ χ)))

Assertion

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

Distinct variable groups:   φ,x   χ,x

Allowed substitution hints:   φ(y)   ψ(x,y)   χ(y)

Proof of Theorem cbvexd

Step	Hyp	Ref	Expression
1		cbvald.1	. . . 4 ⊢ Ⅎyφ
2		cbvald.2	. . . . 5 ⊢ (φ → Ⅎyψ)
3	2	nfnd 1791	. . . 4 ⊢ (φ → Ⅎy ¬ ψ)
4		cbvald.3	. . . . 5 ⊢ (φ → (x = y → (ψ ↔ χ)))
5		notbi 286	. . . . 5 ⊢ ((ψ ↔ χ) ↔ (¬ ψ ↔ ¬ χ))
6	4, 5	syl6ib 217	. . . 4 ⊢ (φ → (x = y → (¬ ψ ↔ ¬ χ)))
7	1, 3, 6	cbvald 2008	. . 3 ⊢ (φ → (∀x ¬ ψ ↔ ∀y ¬ χ))
8	7	notbid 285	. 2 ⊢ (φ → (¬ ∀x ¬ ψ ↔ ¬ ∀y ¬ χ))
9		df-ex 1542	. 2 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
10		df-ex 1542	. 2 ⊢ (∃yχ ↔ ¬ ∀y ¬ χ)
11	8, 9, 10	3bitr4g 279	1 ⊢ (φ → (∃xψ ↔ ∃yχ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cbvexdva  2011  vtoclgft  2906  dfid3  4769