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Theorem eqeq1dALT 2654
 Description: Shorter proof of eqeq1d 2653 based on more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
eqeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
eqeq1dALT (𝜑 → (𝐴 = 𝐶𝐵 = 𝐶))

Proof of Theorem eqeq1dALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqeq1d.1 . . . . . 6 (𝜑𝐴 = 𝐵)
2 dfcleq 2645 . . . . . 6 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylib 208 . . . . 5 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4319.21bi 2097 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
54bibi1d 332 . . 3 (𝜑 → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
65albidv 1889 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝑥𝐶) ↔ ∀𝑥(𝑥𝐵𝑥𝐶)))
7 dfcleq 2645 . 2 (𝐴 = 𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
8 dfcleq 2645 . 2 (𝐵 = 𝐶 ↔ ∀𝑥(𝑥𝐵𝑥𝐶))
96, 7, 83bitr4g 303 1 (𝜑 → (𝐴 = 𝐶𝐵 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521   = wceq 1523   ∈ wcel 2030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644 This theorem is referenced by: (None)
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