Theorem drsb1 2022

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
drsb1	⊢ (∀x x = y → ([z / x]φ ↔ [z / y]φ))

Proof of Theorem drsb1

Step	Hyp	Ref	Expression
1		equequ1 1684	. . . . 5 ⊢ (x = y → (x = z ↔ y = z))
2	1	sps 1754	. . . 4 ⊢ (∀x x = y → (x = z ↔ y = z))
3	2	imbi1d 308	. . 3 ⊢ (∀x x = y → ((x = z → φ) ↔ (y = z → φ)))
4	2	anbi1d 685	. . . 4 ⊢ (∀x x = y → ((x = z ∧ φ) ↔ (y = z ∧ φ)))
5	4	drex1 1967	. . 3 ⊢ (∀x x = y → (∃x(x = z ∧ φ) ↔ ∃y(y = z ∧ φ)))
6	3, 5	anbi12d 691	. 2 ⊢ (∀x x = y → (((x = z → φ) ∧ ∃x(x = z ∧ φ)) ↔ ((y = z → φ) ∧ ∃y(y = z ∧ φ))))
7		df-sb 1649	. 2 ⊢ ([z / x]φ ↔ ((x = z → φ) ∧ ∃x(x = z ∧ φ)))
8		df-sb 1649	. 2 ⊢ ([z / y]φ ↔ ((y = z → φ) ∧ ∃y(y = z ∧ φ)))
9	6, 7, 8	3bitr4g 279	1 ⊢ (∀x x = y → ([z / x]φ ↔ [z / y]φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  [wsb 1648

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  df-sb 1649

This theorem is referenced by:  sbequi  2059  nfsb4t  2080  sbco3  2088  sbcom  2089  sb9i  2094  iotaeq  4348