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Theorem sbequALT 2597
Description: Alternate version of sbequ 2090. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.xz (𝜃 ↔ ((𝑧 = 𝑥𝜑) ∧ ∃𝑧(𝑧 = 𝑥𝜑)))
dfsb1.yz (𝜏 ↔ ((𝑧 = 𝑦𝜑) ∧ ∃𝑧(𝑧 = 𝑦𝜑)))
Assertion
Ref Expression
sbequALT (𝑥 = 𝑦 → (𝜃𝜏))

Proof of Theorem sbequALT
StepHypRef Expression
1 dfsb1.xz . . 3 (𝜃 ↔ ((𝑧 = 𝑥𝜑) ∧ ∃𝑧(𝑧 = 𝑥𝜑)))
2 dfsb1.yz . . 3 (𝜏 ↔ ((𝑧 = 𝑦𝜑) ∧ ∃𝑧(𝑧 = 𝑦𝜑)))
31, 2sbequiALT 2596 . 2 (𝑥 = 𝑦 → (𝜃𝜏))
42, 1sbequiALT 2596 . . 3 (𝑦 = 𝑥 → (𝜏𝜃))
54equcoms 2027 . 2 (𝑥 = 𝑦 → (𝜏𝜃))
63, 5impbid 215 1 (𝑥 = 𝑦 → (𝜃𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  sbco2ALT  2615
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