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Theorem iineq1d 44487
Description: Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypothesis
Ref Expression
iineq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
iineq1d (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem iineq1d
StepHypRef Expression
1 iineq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 iineq1 5017 . 2 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
31, 2syl 17 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   ciin 5001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-ral 3059  df-rex 3068  df-iin 5003
This theorem is referenced by:  iineq12dv  44503  smflimlem2  46189  smflimlem3  46190  smflimlem4  46191  smflim2  46223  smflimsuplem1  46237  smflimsuplem7  46243  smflimsup  46245  smfliminf  46248
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