Theorem drsb1 1772

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 1689	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
2	1	sps 1518	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
3	2	imbi1d 230	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → 𝜑)))
4	2	anbi1d 461	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑧 ∧ 𝜑) ↔ (𝑦 = 𝑧 ∧ 𝜑)))
5	4	drex1 1771	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑)))
6	3, 5	anbi12d 465	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) ↔ ((𝑦 = 𝑧 → 𝜑) ∧ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑))))
7		df-sb 1737	. 2 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ((𝑥 = 𝑧 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)))
8		df-sb 1737	. 2 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ((𝑦 = 𝑧 → 𝜑) ∧ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑)))
9	6, 7, 8	3bitr4g 222	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  ∃wex 1469  [wsb 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-sb 1737

This theorem is referenced by:  sbequi  1812  nfsbxy  1916  nfsbxyt  1917  sbcomxyyz  1946  iotaeq  5104