Theorem del43 856

Description: A distinctor elimination lemma for substitution.

Assertion

Ref	Expression
del43	⊢ (∀x x = y → ([z / x]φ → [z / y]φ))

Proof of Theorem del43

Step	Hyp	Ref	Expression
1		ax-8 798	. . . . . 6 ⊢ (y = x → (y = z → x = z))
2	1	a4s 682	. . . . 5 ⊢ (∀y y = x → (y = z → x = z))
3	2	eq4s 822	. . . 4 ⊢ (∀x x = y → (y = z → x = z))
4	3	syl4d 28	. . 3 ⊢ (∀x x = y → ((x = z → φ) → (y = z → φ)))
5		ax-8 798	. . . . . 6 ⊢ (x = y → (x = z → y = z))
6	5	a4s 682	. . . . 5 ⊢ (∀x x = y → (x = z → y = z))
7	6	anim1d 432	. . . 4 ⊢ (∀x x = y → ((x = z ∧ φ) → (y = z ∧ φ)))
8	7	del40 839	. . 3 ⊢ (∀x x = y → (∃x(x = z ∧ φ) → ∃y(y = z ∧ φ)))
9	4, 8	anim12d 431	. 2 ⊢ (∀x x = y → (((x = z → φ) ∧ ∃x(x = z ∧ φ)) → ((y = z → φ) ∧ ∃y(y = z ∧ φ))))
10		df-sb 853	. 2 ⊢ ([z / x]φ ↔ ((x = z → φ) ∧ ∃x(x = z ∧ φ)))
11		df-sb 853	. 2 ⊢ ([z / y]φ ↔ ((y = z → φ) ∧ ∃y(y = z ∧ φ)))
12	9, 10, 11	3imtr4g 426	1 ⊢ (∀x x = y → ([z / x]φ → [z / y]φ))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852

This theorem is referenced by:  del43b 857  sbequi 876  sb9i 920

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-10 800  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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